Information measures of quantum systems * * * Collective...

Information measures of quantum systems

* * *

Collective Rydberg excitations of an atomic gas confined in

a ring lattice

Tesis Doctoral presentada por

Beatriz M. Olmos Sanchez

Tesis Doctoral dirigida por:

Jesus Sanchez-Dehesa Moreno-Cid

Rosario Gonzalez Ferez

Igor Lesanovsky

Departamento de Fısica Atomica, Molecular y Nuclear

Universidad de Granada

III Dna. Rosario Gonzalez Ferez, Doctora en Ciencias Fısicas y Profesora Titular del Departa-mento de Fısica Atomica, Molecular y Nuclear de la Facultad de Ciencias de la Universidadde Granada, D. Igor Lesanovsky, Doctor in Physics and Lecturer in the School of Physics and Astronomyof the University of Nottingham y D. Jesus Sanchez-Dehesa Moreno-Cid, Doctor en Fısica, Doctor en Matematicas y Catedraticodel Departamento de Fısica Atomica, Molecular y Nuclear de la Facultad de Ciencias de laUniversidad de Granada,

MANIFIESTAN:Que la presente memoria titulada: ”Information measures of quantum systems and collective

Rydberg excitations of an atomic gas confined in a ring lattice”, presentada por Beatriz MontserratOlmos Sanchez para optar al Grado de Doctora en Fısica, ha sido realizada bajo nuestra direccionen el Departamento de Fısica Atomica, Molecular y Nuclear y el Instituto Carlos I de Fısica Teoricay Computacional de la Universidad de Granada,

2 de Febrero de 2010

Fdo.: Rosario Gonzalez Ferez

Fdo.: Igor Lesanovsky

Fdo.: Jesus Sanchez-Dehesa Moreno-Cid

Fdo.: Beatriz Montserrat Olmos Sanchez

V

Tıtulo de Doctor con Mencion Europea

Con el fin de obtener la Mencion Europea en el Tıtulo de Doctor (aprobada en Junta de Gobiernode la Universidad de Granada el 5 de Febrero de 2001), se han cumplido, en lo que atane a estaTesis Doctoral y a su Defensa, los siguientes requisitos:

1. La memoria esta escrita parcialmente en ingles y en espanol.

2. Uno o mas de los miembros del tribunal provienen de una Universidad europea no espanola.

3. Una parte de la defensa se ha realizado en ingles.

4. Una parte de esta Tesis Doctoral se ha realizado en Inglaterra, en la School of Physics andAstronomy de la Universidad de Nottingham.

VII

Agradecimientos/Acknowledgements

Escribir los agradecimientos es comprometido y difıcil, porque se que mucha gente solo se leeraesta pagina de la tesis. Quiero dedicarsela a todos los que han significado algo o mucho para mıdurante estos cuatro anos, ya que la suma de todo es lo que me trae al punto en el que me encuentro.Alla vamos: A mis directores de tesis. A Jesus, por haber depositado tu confianza en mı. Desde el prin-

cipio, cuando decidiste darme una oportunidad de empezar en el mundo de la investigacion,hasta el final, donde has sabido darme la libertad suficiente, aunque siempre pendiente de miavance. A Rosario, por haber sido no solo un gran punto de apoyo para continuar trabajandodurante los ultimos anos, sino tambien una amiga con la que poder charlar siempre que meha hecho falta. To Igor, for being both a patient and a hard supervisor. For having taughtme so much and having asked always for a bit more when you knew I could give it. To the people I met in Innsbruck. Peter Zoller, for giving me the great opportunity of beingthere and learn so much. Maggie and Michael, sharing the office with you both was reallygreat, talking about fashion and/or video games. Markus and Sebastian, contando chistes anddancing salsa. Gemma, bailando rock and roll mientras hablabamos catalan (vale, hablabas,jejeje). Claudiu, for all the chats on the terrace. You all made me feel sad when I had toleave, and make me remember those five months with a smile, see you soon! To the people in Nottingham. Peter, Thomas, Olga, Lucia, Anton, Sonali, Ivette, Celine,Marta and the rest, because you all make Nottingham feel a bit more like home. A Sandra, Su y Carmen, a mis antonias. Por lo que nos hemos reıdo y llorado juntas estosanos, que cumplamos muchos mas. A Dani, Jorge, Pablo, Antonio, Alex, (no me mareeis con los ordenes alfabeticos o no, eh?)por las noches frikis con cerveza y Big Bang Theory, El Nucleo y Torque, por Naruto yNarute, porque ahora digo con orgullo: !yo tambien soy friki! Jesus, por lo anterior y mas,por no dejar que me rindiera. A Paqui y Lidia, por aullarle a la luna conmigo. Por haber hecho de nuestra ultima temporadaen Granada (al menos de momento), algo memorable. Estoy a la espera de las coordenadasde la proxima. A Pili y Rake, por seguir ahı. Por ser alguien con quien poder contar, pase lo que pase; por locomoda que me siento con vosotras aunque no nos veamos tan a menudo como debieramos.Ha sido y seguira siendo un placer crecer y veros crecer al mismo tiempo. Y Pili, a ver si nosdan el Nobel, no? A Sheila, que ha andado el camino conmigo desde que eramos companeras de practicas hastaque nos ha tocado corregirlas, y ha aguantado conmigo que la gente todavıa no sepa quienes Sheila y quien Bea (deberıamos ponernos identificacion como en los congresos. . . ). Poravisarme de los plazos, sin ti habrıa perdido la beca por no echar un papel, fijo. Por ser miamiga, por ser unica. A mi hermano, por haber hecho de mi ninez un juego y haberse convertido como quien noquiere la cosa en una persona maravillosamente diferente a mı de la que me siento orgullosasiempre.

VIII A mis padres. ¿Por que? Bueno, porque soy todo lo que vosotros me habeis dado. Por haberpermitido, pacientes, que cometiera errores y aprendiera de ellos, por animarme y ayudarme.Porque siempre me habeis hecho ver los malos momentos con mejor cara, ya sea con un buenconsejo o un buen abrazo. Por decir que soy una llorona porque soy muy inteligente. En fin,por estar ahı, y darme la certeza de que tengo un punto apoyo constante e incondicional, queno es poco. Os quiero muchısimo. Y a Igor, por todo lo que has sido, eres y seras.

Contents

Foreword XIII

Prologo XV

I. Information measures of quantum systems 1

1. Introduction 3

2. Fisher information of D-dimensional hydrogenic systems 7

2.1. The hydrogenic problem in D dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1. D-dimensional central potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.2. Electron distribution in position space . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.3. Electron distribution in momentum space . . . . . . . . . . . . . . . . . . . . . . 11

2.2. The Fisher information in position space . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1. Radial contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.2. Angular contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3. The Fisher information in momentum space . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1. Radial contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4. Alternative method and application to the case D=3 . . . . . . . . . . . . . . . . . . . 18

2.5. Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3. Parameter-based Fisher Information of orthogonal polynomials 21

3.1. Some properties of the parameter-dependent classical orthogonal polynomials . . . . 22

3.2. Parameter-based Fisher information of Jacobi and Laguerre polynomials . . . . . . . 24

3.3. Fisher information of Gegenbauer and Grosjean polynomials . . . . . . . . . . . . . . . 26


II. Collective Rydberg excitations of an atomic gas confined in a ring lattice 33

4. Introduction 35

5. Preliminary concepts 37

5.1. Interaction of a two-level atom and a monochromatic light field . . . . . . . . . . . . . 37

5.2. Second quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.3. Rydberg atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.4. Interacting Rydberg atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.4.1. Simple model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.4.2. Rydberg blockade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

X Contents

6. Laser-driven atomic gas confined to a ring lattice 45

6.1. The system and its Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.2. Analysis of the symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7. Strong interaction: Perfect blockade 49

7.1. Perfect blockade regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

7.2. Graph: general behavior of expectation values of observables . . . . . . . . . . . . . . 50

7.3. Numerical results: time evolution of expectation values of observables . . . . . . . . . 51

7.3.1. Two-sites density matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

7.3.2. Rydberg density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

7.3.3. Density-density correlation function . . . . . . . . . . . . . . . . . . . . . . . . . 54

7.3.4. Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.4. Non-perfect Blockade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58


8. Thermalization of a strongly interacting 1D Rydberg lattice gas 63

8.1. Hamiltonian in excitation number space . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

8.2. Time evolution in the excitation number subspace . . . . . . . . . . . . . . . . . . . . . 64

8.2.1. Time evolution of the projection operators . . . . . . . . . . . . . . . . . . . . . 64

8.2.2. Effective equation of motion for ρm . . . . . . . . . . . . . . . . . . . . . . . . . 65

8.3. The steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

8.4. Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

8.4.1. Evolution into the steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

8.4.2. The steady state and its dependence on the initial condition . . . . . . . . . . 70

8.4.3. Reduced density matrix in excitation number space . . . . . . . . . . . . . . . . 71

8.4.4. Connection with the microcanonical ensemble . . . . . . . . . . . . . . . . . . . 71


9. Strong laser driving: fermionic collective excitations 73

9.1. Constrained dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

9.2. Jordan-Wigner transformation on a ring . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

9.3. Many-body states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

9.3.1. Fully-symmetric states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

9.3.2. Energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

9.3.3. Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

9.4. Excitation of many particle states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

9.4.1. Direct trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

9.4.2. Excitation from the ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

9.5. Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

9.5.1. Quench of a superfluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87


10.Summary and outlook 91

11.Conclusiones 93

A. Excitation to a Rydberg state by a two-photon transition 95

B. Perfect blockade beyond the nearest neighbor 97

C. Perturbative corrections to the xy-model 99

Contents XI

D. Numerical methods 101

D.1. Necklaces and bracelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101D.2. Matrix representation of the Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 102

XII Contents

Foreword

This thesis encompasses two parts. In the first one, which is called ’Information measures ofquantum systems’, we present a contribution to the information theory of quantum mechanicalsystems and theory of special functions. In the second part, entitled ’Collective Rydberg excitationsof an atomic gas confined in a ring lattice’, we move to a different topic situated in the realm ofultracold atoms, Rydberg gases and complex many-body systems. In the following we brieflyoutline the contents of the thesis. A more detailed introduction and motivation can be found atthe beginning of the respective parts.

In Part I, the Fisher information, which is a local information-theoretic measure, is both mathe-matically and physically studied. We do this by focussing on two applications which explicitly showthose two aspects of the previous measure. Firstly, we determine the (translationally invariant)Fisher information of the multidimensional hydrogenic systems in both position and momentumspaces. To do so, we exploit the algebraic properties of the special functions which control the wave-functions of the quantum states of these systems; namely, the Laguerre and Gegenbauer polynomialsand the hyperspherical harmonics. The results are explicitly given in terms of the dimensional pa-rameter and the hyperquantum numbers of the state. Secondly, the role of the Fisher informationin estimation theory is highlighted. In particular, the parameter-based Fisher information the or-thogonal polynomials Jacobi, Laguerre, Gegenbauer and Grosjean are analytically derived in termsof the polynomial degree and the parameter(s) which characterize them.

In Part II, we study a gas of ultracold Rydberg atoms trapped in a ring lattice and analyzethoroughly its excitation properties. First, the system is studied in the perfect blockade regime,that is, the double excitation of neighboring sites of the ring is neglected. In this framework, we firstconsider the time evolution of local properties, like the density of Rydberg states, the correlationsand the entanglement, by means of the numerical solution of the Schrodinger equation. We find thatthe expectation values of certain observables, for example the number of Rydberg atoms, equilibrateafter a short period of time, that is, they acquire a constant value with small fluctuations. Theorigin of this thermalization of a closed quantum system and its relation to the microcanonicalensemble is then further investigated. Moreover, we focus on a regime where the laser field is muchstronger than the interaction between Rydberg atoms. Here, we find that the system is analyticallysolvable and allows us to access entangled many-particle states. We characterize these states bymeans of, for example, their correlations and outline a possible experimental route towards theirexcitation.

XIV Contents

Prologo

Esta memoria de tesis doctoral consta de dos partes. En la primera, titulada ’Medidas de infor-macion de sistemas cuanticos’, se presenta una contribucion a la teorıa de informacion de sistemasmecano-cuanticos y a la teorıa de funciones especiales. La segunda parte del trabajo, bajo el tıtulode ’Excitaciones Rydberg colectivas de un gas atomico confinado en una red circular’, esta enmar-cada en el campo de la fısica de atomos ultrafrıos, estados Rydberg y sistemas multiparticularescomplejos. A continuacion se describen brevemente los contenidos de esta memoria de tesis. Alprincipio de cada una de las partes, se presentara una introduccion y motivacion mas detalladas delas mismas.

En la Parte I, se estudia la informacion de Fisher, que es una medida teorica de informacionde caracter local. Este objetivo se aborda desde un punto de vista tanto matematico como fısico.En primer lugar, se determina la informacion de Fisher (traslacionalmente invariante) de los sis-temas hidrogenoides multidimensionales tanto en el espacio de posiciones como en el de momentos.Para ello, se hace un uso exhaustivo de las propiedades algebraicas de las funciones especiales quecontrolan las funciones de onda de los estados cuanticos de estos sistemas, es decir, los polinomiosde Laguerre y Gegenbauer junto con los armonicos hiperesfericos. El resultado obtenido dependeexplıcitamente de la dimension y de los numeros cuanticos del correspondiente estado. A contin-uacion, se pone de manifiesto el papel que juega la informacion de Fisher en la teorıa de estimacionde parametros. En particular, se deriva analıticamente la informacion de Fisher con respecto alparametro que caracteriza los polinomios ortogonales de Jacobi, Laguerre, Gegenbauer y Grosjean.

En la Parte II, se analizan las propiedades espectrales y dinamicas de un gas de atomos Rydbergconfinado en una red optica circular monodimensional excitados mediante un campo laser. Esteestudio se comienza abordando el regimen de bloqueo perfecto, donde se considera prohibida laexcitacion simultanea de dos atomos situados en pozos contiguos de la red. En este caso, se investigala evolucion temporal de varias magnitudes fısicas, tales como la densidad de estados Rydberg, lacorrelacion y el entrelazamiento, mediante la resolucion numerica de la ecuacion Schrodinger. Esteregimen se caracteriza por el estado de equilibrio que alcanzan algunos de los valores esperados delos observables alrededor del cual muestran pequenas fluctuaciones. Se ha hecho especial hincapieen el analisis del origen de la termalizacion de este sistema cuantico aislado, investigando su relacioncon el conjunto microcanonico. A continuacion, se considera el regimen dominado por la interaccioncon el campo laser. Se muestra que en este caso el sistema es analıticamente resoluble, y permitela creacion de estados multiparticulares entrelazados. Se han estudiado las propiedades de dichosestados, y propuesto un esquema experimental para excitarlos.

XVI Contents

Part I.

Information measures of quantum

systems

1. Introduction

In the last few years, the application of information-theoretic ideas and techniques to the studyof multielectronic systems has proved to be an interesting new channel of cross-fertilization be-tween atomic and molecular physics, D-dimensional physics and theory of information [1, 2, 3, 4,5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. This line of research has been addressed to the most rele-vant measures of information (Renyi and Shannon entropies, Fisher information. . . ) for severalquantum-mechanical potentials of a specific form [16, 17, 18] such as hydrogenic systems with stan-dard and non-standard dimensionalities [19, 20], circular membranes [21], confined systems [22, 23]and, in general, many-electron systems [14, 24, 25, 26, 27]. However, the analytic determination ofthese information-theoretic measures has not yet been possible even for the single-particle systemswith an analytically solvable Schrodinger equation, whose corresponding wavefunctions describingtheir physical states are controlled by special functions of mathematical physics (classical orthog-onal polynomials, spherical harmonics, Bessel functions. . . ). This is due to the lack of knowledgeof the information-theoretic properties of special functions despite of the many results provided bythe theory of the orthogonal polynomials and potential theory [19, 28, 29, 30, 31, 32].

The Fisher information is a gradient functional of the involved probability density, that is, it hasthe property of locality in the sense that it is very sensitive to irregularities of the distribution. Thisis not the case for the Renyi and Shannon entropies, which are power and logarithmic functionals ofthe probability density. This part of the thesis is mainly focused on the analytical determination ofthe Fisher information of the D-dimensional single-particle systems subject to Coulomb potentials.

The Fisher information was originally introduced in 1925 by R.A. Fisher in the theory of sta-tistical estimation [33]. In a system, let us consider the problem of estimating a parameter withdefinite but unknown value θ by means of N measurements y ≡ y1, . . . yN. The data obtainedfluctuate around the real value θ with a noise obeying a random noise distribution, so that the setof measurements can be given as

y = θ + x,where x ≡ x1, . . . xN are the values of the noise. These data are used to form an estimate θ(y)of the parameter θ, and the whole measurement process is called to be ’smart’ when the estimateobtained by this procedure is closer to the real value of the parameter than any of the data containedin y. In this framework, the Fisher information arises as a measure of the expected error estimatinga parameter in a smart measurement, i.e., from an imperfect observation in the presence of somerandom noise.

The system is specified by a likelihood or conditional probability law given by the family ofprobability densities ρθ(y) ≡ ρ(y ∣ θ). The Fisher information of the measurement process governedby ρθ(y) is defined as [33]

I(θ) ∶= ∫ [∂ lnρθ(y)∂θ

]2 ρθ(y)dy = ∫ [∂ρθ(y)

∂θ]2

ρθ(y) dy, (1.1)

where the integrals are over all the corresponding space. It can be formally shown from thisdefinition and the mean-square error of the estimate θ(y)

σ2(θ) ≡ ∫ [θ(y) − θ]2 ρθ(y)dy,

4 Introduction

that they accomplish the so-called Cramer-Rao inequality [34]

σ2(θ)I(θ) ≥ 1. (1.2)

Hence, the Fisher information is formally seen to be a measure of the ability to estimate theparameter θ, i.e., it gives the minimum error in estimating θ given the probability density ρθ(y).

Note that the Fisher information (1.1) explicitly depends, in general, upon the parameter θ.There is an important exception to this rule. For the sake of simplicity, let us assume that thereis only one measurement, i.e., N = 1. In the particular case where the noise fluctuations x areindependent of the size of θ, i.e., shift invariance, the probability density accomplishes

ρθ(y) = ρ(y − θ) = ρ(x),and the equation (1.1) reduces to

I = ∫ [∂ρ(x)∂x]2

ρ(x) dx. (1.3)

Here, the Fisher information is a functional that gives a measure of the amount of gradient andthat is sensitive to local changes of the probability distribution. This definition of the so-calledtranslationally-invariant Fisher information is important since in fact a vast number of phys-ical systems obey this shift invariance. In particular, this property holds for any isolated quantummechanical system subject to a central potential centered in θ.

Nowadays, the Fisher Information is being used in numerous scientific areas ranging from statis-tics [33, 35, 36], information theory [34] and signal analysis [37], to quantum physics [6, 11]. Thismeasure is also the main theoretic tool of the Extreme Physical Information principle (EPI), ageneral variational principle which allows to derive numerous fundamental equations of physics[4, 5, 6, 11, 15], such as Maxwell, Dirac, Klein-Gordon and Schrodinger equations, as well as ther-modynamics’ first law among others. It has been also shown that using the EPI leads to obtain aswell the non-equilibrium thermodynamics [9].

This information-theoretic quantity has, among other characteristics, a number of importantproperties beyond the mere non-negativity, which deserve to be resembled here:

1. Additivity for independent events. If ρθ(y, z) = ρ(1)θ(y) ⋅ ρ(2)

θ(z), then

I [ρθ(y, z)] = I [ρ(1)θ(y)] ⋅ I [ρ(2)

θ(z)]

2. Scaling invariance. The Fisher information is invariant under sufficient transformations y =t(z), so that

I [ρθ(y)] = I [ρθ(z)] .This property is not only closely related to Fisher’s maximum likelihood method [38] but alsoit is very important for the theory of statistical inference.

3. The previously cited Cramer-Rao inequality [34]. It states that the reciprocal of the Fisherinformation I(θ) bounds from below the mean square error of an unbiased estimator f ≡ θ(y)of the parameter θ; i.e.,

σ2(f) ≥ 1

I(θ) ,where σ2(f) denotes the variance of f . This inequality, which lies at the heart of statisticalestimation theory, shows how much information the distribution provides about a parameter.

5

4. Relation to other information-theoretic properties. The Fisher information is related to theso-called Shannon entropy [39] of the probability distribution ρθ(y) defined as

S(ρθ) ∶= −∫ ρθ(y) ln ρθ(y)dy, (1.4)

via the elegant de Bruijn’s identity [34, 40, 41]. More precisely, let ρθ be the convolutionprobability density of any probability density ρ with the normal density with zero mean andvariance θ > 0. Then, the following relation stands,

∂

∂θS(ρθ) = 1

2I(ρθ).

Moreover, the Fisher information I(ρθ) satisfies, under proper regularity conditions, the lim-iting property [6, 11, 42]

I(ρθ) = limǫ→0

2

ǫ2D (ρθ+ǫ∥ρθ) , (1.5)

where

D(p∥q) ∶= ∫ p(y) ln p(y)q(y)dy (1.6)

denotes the relative or cross entropy (Kullback-Leibler divergence) of the probability den-sity p(y) and a reference distribution q(y). There exist further connections of the Fisherinformation with other information-theoretic properties, as well as with expectation values oflogarithmic and power functions of the probability distribution; see, e.g., [2, 43, 44].

5. Applications in quantum physics. The Fisher information I(ρθ) plays a fundamental role inthe quantum-mechanical description of physical systems. It has been shown

a) to be a measure of both disorder and uncertainty of a probability distribution, as thor-oughly discussed in the Refs. [4, 6, 11];

b) to be a measure of nonclassicality for quantum systems [7, 8];

c) to describe, some factor apart, various macroscopic quantities such as the kinetic [1, 41]and the Weiszacker energies [2, 3];

d) to derive numerous fundamental equations in physics, such as the Schrodinger and Klein-Gordon equations of motion [5, 11] as well as the Euler equation of the density functionaltheory [10], from the principle of minimum Fisher information, in a similar way as theShannon entropy is the starting point of the maximum entropy methods;

e) to predict some non-linear spectral phenomena, such as the avoided crossings of energylevels encountered in atomic and molecular systems under strong external electric andmagnetic fields [13], and correlations in many-electron systems [12];

f) to be involved in numerous uncertainty inequalities [40, 41, 45, 46];

g) to detect the transition state and the stationary points of a chemical reaction, as wellas the bond breaking/forming regions of elementary ones [27].

In this part of the thesis, we present our contributions to the present knowledge on this infor-mation quantity in two chapters. In Chapter 2, we calculate the so-called translationally-invariantFisher information of an important quantum system: the D-dimensional hydrogenic atom. Weperform this task by a direct application of the definition given in (1.3) to the probability distri-bution obtained squaring the wavefunction that describes the dynamics of the system. We obtainthe corresponding results in a closed analytical form (for both position and momentum spaces)and find that they can be expressed in terms of the hyperquantum numbers, which describe the

6 Introduction

corresponding energy eigenstate. In a second stage (Chapter 3), we perform the fundamental taskof calculating the Fisher information of the classical families of polynomials with respect to theparameters that characterize them. In particular, we treat the Laguerre, Jacobi, Gegenbauer andGrosjean polynomials and express the results in terms of their degree and corresponding parameter.At the end of each of these two chapters, we analyze and discuss the results obtained. The researchcarried out in this part has given as a result the following publications: J.S. Dehesa, S. Lopez-Rosa, B. Olmos and R.J. Yanez, ”Information measures of hydrogenic

systems, Laguerre polynomials and spherical harmonics”, J. Comput. Appl. Math., vol. 179,p. 185, 2005. J.S. Dehesa, S. Lopez-Rosa, B. Olmos and R.J. Yanez, ”The Fisher information of D-dimensional hydrogenic systems in position and momentum spaces”, J. Math. Phys., vol.47, p. 052104, 2006. J.S. Dehesa, B. Olmos and R.J. Yanez, ”Parameter-based Fisher information of orthogonalpolynomials”, J. Comput. Appl. Math., vol. 214, p. 136, 2008.

2. Fisher information of D-dimensional

hydrogenic systems

The quantum state of a many-particle system is in general described by the physical solution Ψ(r)of the time-independent Schrodinger equation

HΨ(r) = EΨ(r),where H is the Hamiltonian (independent of time) that drives the dynamics and E is the energyof the corresponding state. The spatial distribution of the system is then given by the probabilitydensity P (r) = ∣Ψ(r)∣2, so that most of the physical information of the state is contained in thisprobability distribution.

One can characterize the information-theoretic content of a general probability distribution P (r)by means of two different and complementary quantities among others: the Shannon entropy andthe Fisher information. The Shannon entropy [39, 47] is a logarithmic functional of the probabilitydensity (1.4). It has been shown that it is a global measure of the smoothness (or disorder) of P (r).On the other hand, the so-called translationally-invariant Fisher information (we will refer to it inthis chapter simply as Fisher information) is defined as the gradient functional of the density

I = ∫ [∇P (r)]2P (r) dr, (2.1)

so it has the property of locality because it is sensitive to local rearrangements of the positionvariable r. I is actually a measure of the degree of disorder of the system described by the cor-responding probability distribution P (r). The higher this quantity is, the more concentrated isthe single-particle density (more gradient content), the smaller the uncertainty and the higher theaccuracy is when predicting the localization of the particle (see Fig. 2.1).

However, the analytical determination of the Fisher information from first principles is not asimple task, not even for single-particle systems. In this chapter we shall perform the analyticalcalculation of the position and momentum Fisher information of a known and very importantquantum system: the D-dimensional hydrogenic atom. So, the momentum hydrogenic orbitals (i.e.,the solution to the non-relativistic, time-independent Schrodinger equation in momentum space)will also be studied. This is not only because the momentum eigenstates have usually been avoided

r r

I <I1 2

1 2

Figure 2.1.: The translationally-invariant Fisher information gives a measure of the gradient content of aprobability distribution, i.e., the more concentrated the distribution, the bigger the Fisher information.

8 Fisher information of D-dimensional hydrogenic systems

in favor of the position ones up until recently, but also because of (i) their conceptual importance,(ii) the distribution of momenta for real (D=3) atomic systems and specifically for hydrogen atom[48] is nowadays experimentally accessible in atomic, molecular and nuclear experiments, speciallysince the advance of the modern spectroscopy, (iii) the scattering phenomena are more convenientlyviewed in numerical simulations by means of momentum space (see, e.g., [49]) and (iv) they playa very relevant role in numerous other physical processes with atoms and molecules which aregoverned by simple functions of the momentum transfer [50].

The chapter is structured as follows. First, the known position and momentum wavefunctions ofthe hydrogenic system in D dimensions are described in detail, and the corresponding probabilitydensities are explicitly shown. Then, we find closed expressions for the Fisher information of thehydrogenic eigenstates in terms of the D quantum numbers which characterize them in positionand momentum spaces, respectively. Finally, some concluding remarks are given.

2.1. The hydrogenic problem in D dimensions

In this section we fix our notation and we describe in hyperspherical coordinates the wavefunc-tions of the D-dimensional hydrogenic eigenstates, i.e., the solutions to the non-relativistic, time-independent Schrodinger equation inD dimensions. The configuration (or position) and momentumspaces are considered, and the associated probability distributions are obtained.

The position and momentum hydrogenic eigenstates in D-dimensions are known to be conformedin polar coordinates by an angular and a radial part. The angular part, which is common toboth cases, is given in terms of the hyperspherical harmonics because of the radially symmetriccharacter of the Coulomb potential. The radial part is controlled by the Laguerre and Gegenbauerpolynomials in the position and momentum representations, respectively.

2.1.1. D-dimensional central potential

The D-dimensional time-independent Schrodinger equation for a particle moving in a central po-tential VD(r) reads (in atomic units)

(− 1

2µ∇2

D + VD(r))Ψ (r) = EΨ (r) , (2.2)

where µ is the reduced mass of the system and ∇2D stands for the Laplacian operator in D di-

mensions. Due to the symmetry of the problem, we work in hyperspherical coordinates r ≡(r, θ1, . . . , θD−1) ≡ (r,ΩD−1) related to the Cartesian set (x1, . . . , xD) asx1 = r sin θ1 sin θ2 . . . sin θD−2 cos θD−1x2 = r sin θ1 sin θ2 . . . sin θD−2 sin θD−1x3 = r sin θ1 sinθ2 . . . cos θD−2

⋮xD−1 = r sinθ1 cos θ2

xD = r cos θ1,with 0 ≤ θi ≤ π, i = 1, ...,D−2, 0 ≤ θD−1 ≤ 2π for D ≥ 2 and such that ∑D

i=1 x2i = r2. In this framework,

the gradient operator ∇D reads

∇D = ∂

∂rr + 1

r

D−1

∑i=1

1

∏i−1k=1 sin θk

∂

∂θiθi, (2.3)

2.1 The hydrogenic problem in D dimensions 9

so the Laplacian operator yields

∇2D = 1

rD−1∂

∂rrD−1

∂

∂r− Λ2

r2, (2.4)

with Λ being the generalized angular momentum in D dimensions, that depends on the D − 1hyperangular coordinates ΩD−1 [51, 52]. Λ2 then reads

Λ2 = −D−1

∑i=1

(sin θi)i−D+1∏i−1

k=1 sin θk2

∂

∂θi[(sin θi)D−1−i ∂

∂θi] ,

and satisfies the eigenvalue equation

Λ2Yν(ΩD−1) = l(l +D − 2)Yν(ΩD−1). (2.5)

Here, Yν(ΩD−1) stand for the hyperspherical harmonics [51], that depend on the D − 1 quantumnumbers ν ≡ (l ≡ ν1, ν2, . . . , νD−1) corresponding to the respective hyperangular coordinates.These numbers can take integer values accomplishing the constraints l = 0,1, . . . and l ≡ ν1 ≥ ν2 ≥⋅ ⋅ ⋅ ≥ νD−2 ≥ ∣νD−1∣ ≥ 0. Using the so-called partial harmonics Y

(i)νi,νi+1(θi), that are solutions to

− 1

(sin θi)D−1−i∂

∂θi[(sin θi)D−1−i ∂

∂θiY (i)νi,νi+1

(θi)] (2.6)

= [νi (νi +D − i − 1) − νi+1 (νi+1 +D − i − 2)sin2 θi

]Y (i)νi,νi+1(θi),

for i = 1, . . . ,D − 2, the hyperspherical harmonics can be written as

Yν(ΩD−1) = eimθD−1√2π

D−2

∏k=1

Y (k)νk,νk+1(θk). (2.7)

One can easily see, taking into account the normalization of these partial harmonics

∫π

0[Y (i)νi,νi+1

(θi)]2 (sin θi)D−i−1dθi = 1, (2.8)

that the corresponding orthonormality relation for the hyperspherical harmonics reads

∫ΩD−1

dΩD−1Yν(ΩD−1)Yν′(ΩD−1) = δνν′, (2.9)

where the integral is over the generalized solid angle ΩD−1 with volume element

dΩD−1 = (D−2∏i=1

(sin θi)D−1−idθi)dθD−1. (2.10)

Coming back to our initial problem of solving the Schrodinger equation (2.2) and substitutingthe multidimensional Laplacian operator (2.4) into it, one has

[− 1

2µ( ∂2∂r2+ D − 1

r

∂

∂r− Λ2

r2) + VD(r)]Ψ (r) = EΨ (r) .

This equation can be simplified by means of the Ansatz

ΨE,ν (r) = RE,l(r)Yν(ΩD−1).Thus, taking into account the eigenvalue equation (2.5), the radial part RE,l(r) satisfies the radialSchrodinger equation

[− 1

2µ( ∂2∂r2+ D − 1

r

∂

∂r− l(l +D − 2)

r2) + VD(r)]RE,l(r) = ERE,l(r). (2.11)

So far, the only assumption we have made is that the potential is central, i.e., it only depends onthe radial coordinate. Let us now focus on the Coulomb potential.


2.1.2. Electron distribution in position space

Substituting the Coulomb central potential in equation (2.11), i.e., VD(r) = −Zrwith Z being the

atomic number of the atom, it yields

[− 1

2µ( ∂2∂r2+ D − 1

r

∂

∂r− l(l +D − 2)

r2) − Z

r]RE,l(r) = ERE,l(r). (2.12)

Given the structure of the equation, it is convenient to introduce the reduced radial functions

uE,l(r) = rD−12 RE,l(r) in order to eliminate the first derivative, obtaining

[ ∂2∂r2− L(L + 1)

r2+ 2µZ

r]uE,L(r) = −2µEuE,L(r), (2.13)

where we have used the notation for the grand orbital quantum number

L = l + D − 32

.

With the introduction of this generalized quantum number L, the resulting equation reads exactlylike the 3D hydrogenic atom with L taking the place of the orbital quantum number (note that,in fact, L(D = 3) = l). Thus, we proceed in a similar way to the one used to solve the standardhydrogen atom, starting by making the following changes of variable and notation

E = −Z2µ

2η2ρ = 2µZ

ηr,

so that the equation (2.13) reads

[ ∂2∂ρ2− L(L + 1)

ρ2+ ηρ− 1

4]uη,L(ρ) = 0. (2.14)

We consider now the asymptotic behavior of the equation. The solution when ρ→ 0 yields uη,L(ρ) ∼ρL+1 and when ρ → ∞, uη,L(ρ) ∼ e−ρ/2. With this, one finds that the solutions E,R(r) to theeigenvalue problem (2.12) are

Eη = −Z2µ

2η2Rη,L =Nη,L

√ω2L+1

ρD−2L(2L+1)η−L−1 (ρ), (2.15)

where ωα(ρ) = ραe−ρ stands for the weight function with respect to which the Laguerre polynomials

L(α)n (ρ) are orthogonal, i.e.,

∫∞

0L(α)n (ρ)L(α)m (ρ)ωα(ρ) = Γ(n +α + 1)

n!δnm. (2.16)

Moreover, η is related to the principal hyperquantum number n and the dimension as

η = n + D − 32

,

and Nη,L is the normalization constant. The latter is found imposing the normalization condition

∫∞

0∣Rη,L(r)∣2 rD−1dr = 1,

and its expression is given by

Nη,L = (2µZη)D/2 ( (η −L − 1)!

2η [(η +L)!]3)1/2

.

2.1 The hydrogenic problem in D dimensions 11

Summarizing, the eigenfunction of the D-dimensional hydrogenic atom is

Ψη,L,ν(r) = Rη,L(ρ)Yν(ΩD−1) =Nη,L

√ω2L+1

ρD−2L(2L+1)η−L−1 (ρ)Yν(ΩD−1), (2.17)

so that the normalized probability distribution of the system in the position space P (r) = ∣Ψ(r)∣2has the form

P (r) =N 2η,L

ω2L+1(ρ)ρD−2

∣L(2L+1)η−L−1 (ρ)∣2 ∣Yν(ΩD−1)∣2 . (2.18)

Note that for D = 3 we recover the well-known expression of the standard hydrogenic atom.

2.1.3. Electron distribution in momentum space

The wavefunctions in position and momentum spaces are related by means of the D-dimensionalFourier transform [51, 53], defined as

ψ(x) = 1(2π)D/2 ∫ dpe−ip⋅xφ(p)φ(p) = 1(2π)D/2 ∫ dxeip⋅xψ(x),

with p ≡ (p1, . . . , pD) being the Cartesian momentum coordinates. It can be shown (see Ref.[51]) that if the wavefunction in position space is separable in radial and angular coordinatesand, in particular, the angular part is described by the hyperspherical harmonics, i.e., ψ(x) =R(r)Yν(ΩD−1), then the corresponding wavefunction in the momentum space is given by

φ(p) = Sl(p)Yν(ΩpD−1).

In this expression, p ≡ (p, θp1, . . . , θpD−1) ≡ (p,ΩpD−1) are the hyperspherical coordinates in the mo-

mentum space and l ≡ ν1. The transformed radial part can be obtained from R(r) asSl(p) = il

pD/2−1∫∞

0drrD/2R(r)JD/2−l−1(pr),

with Jα(z) being the Bessel functions.Applying this result, one obtains that the D-dimensional hydrogen eigenfunction in momentum

space Φη,L,ν(p) has the form

Φη,L,ν(p) =Mη,L(p)Yν(ΩpD−1), (2.19)

with the radial momentum eigenfunction reading

Mη,L(p) = Kη,L(ηp)L−D−3

2(1 + η2p2)L+2CL+1η−L−1 (1 − η2p21 + η2p2) ,

where p = pZµ

, Cλn(z) are the Gegenbauer polynomials and the normalization constant Kη,L is

Kη,L = ( ηZµ)D/2 22L+3 (η(η −L − 1)!(L!)2

2π(η +L)! )1/2 .Thus, the corresponding probability distribution in momentum space γ(p) = ∣Φη,L,ν(p)∣2 is ex-pressed as

γ(p) = K2η,L

(ηp)2L−D+3(1 + η2p2)2L+4 ∣CL+1η−L−1 (1 − η2p21 + η2p2)∣

2 ∣Yν(ΩpD−1)∣2 . (2.20)

Here again, for D=3 this expression reduces to the corresponding momentum density of the three-dimensional hydrogenic atom [54].


2.2. The Fisher information in position space

In this section we calculate the Fisher information of a general D-dimensional hydrogenic systemin position space. To do so, we apply the extension to the D-dimensional case of the translationallyinvariant Fisher information given in Eq. (2.1), i.e.,

Ir(D) = ∫RD

[∇DP (r)]2P (r) dr, (2.21)

where the volume element in the D-dimensional space is dr = rD−1drdΩD−1. Taking into accountthe form of the hyperspherical harmonics (2.7) and its dependence with the angular coordinateθD−1, one can easily see that the probability distribution P (r) is independent of this coordinate,i.e.,

P (r) = ∣Ψη,L,ν(r, θ1, . . . , θD−2, θD−1)∣2 = [Ψη,L,ν(r, θ1, . . . , θD−2,0)]2 . (2.22)

As a consequence, the equation (2.21) can be rewritten as

Ir(D) = 4∫RD[∇DΨη,L,ν(r, θ1, . . . , θD−2,0)]2 dr. (2.23)

We now make use of the separation of radial and angular coordinates and plug in the explicitexpression of the multidimensional gradient (2.3) to obtain as a result

Ir(D) = IR(D) + 4 ⟨r−2⟩ IY(D),with

IR(D) = 4∫∞

0[ ∂∂rR(r)]2 rD−1dr (2.24)

⟨r−2⟩ = ∫ ∞

0r−2R2(r)rD−1dr

IY(D) = D−2

∑i=1∫ΩD−1

[ 1

∏i−1k=1 sin θk

∂

∂θiYν(θ1, . . . , θD−2,0)]2 dΩD−1, (2.25)

where, for the sake of simplicity, we note R(r) ≡ Rη,L(r). Here, we have used the orthonormalizationcondition (2.9) of the hyperspherical harmonics and the definition of the expectation value of ageneral radial function f(r)

⟨f(r)⟩ = ∫RD

f(r)P (r)dr = ∫ ∞

0f(r)R2(r)rD−1dr, (2.26)

for f(r) = r−2. In the following two subsections, we are going to show that the radial integral IR(D)has the value

IR(D) = (2Zµ)2η3

η − 2

2L + 1 [L(L + 1) − 1

4(D − 1)(D − 3)] (2.27)

and that the angular component IY(D) is given by

IY(D) = L(L + 1) − (D − 1)(D − 3)4

− ∣νD−1∣(2L + 1)2

. (2.28)

Since the expectation value ⟨r−2⟩ reads⟨r−2⟩ = (2Zµ)2

2η31

2L + 1 , (2.29)

2.2 The Fisher information in position space 13

02

46

8

24

68

10

0

0.5

1

|νD−1

|η

a

0 10 20 300

0.02

0.04

|νD−1

|

I r

b

10 20 300

200

400

600

η

I r−1

c

Increasing |νD−1

|

Increasing η

Figure 2.2.: a: Fisher information of the D-dimensional hydrogen atom (Z = 1) varying the grand principaland magnetic quantum numbers η and ∣νD−1∣, respectively. Note that for each value of η = 2,3, . . . we considerall the possible values ∣νD−1∣ = 0,1, . . . , η−1. b: For several fixed values of η, there is a linear variation of theFisher information with the absolute value of νD−1. c: For different ∣νD−1∣, the dependence of the inverseof Fisher information with the principal quantum number is depicted. For νD−1 = 0, the dependence is ofthe form η2, while for νD−1 ≠ 0 there exists a minimum (maximum of the Fisher information) for a value ofη ≠ 1.one finds the following simple expression for the Fisher information of the D-dimensional hydrogenicsystem in the configuration space:

Ir(D) = (2Zµ)2η3

(η − ∣νD−1∣) , (2.30)

valid for D ≥ 2. Remark that it depends only of the grand principal quantum number η andthe grand magnetic quantum number νD−1. It is observed that the Fisher information, as theionization energy, behaves as η−2 with respect to the generalized principal quantum number ηwhen νD−1 = 0. For values of ∣νD−1∣ different from 0, the dependence with the quantum number ηis more complicated. The Fisher information decreases proportionally with the absolute value ofνD−1, which corresponds to the magnetic quantum number m in the three-dimensional case. Allthese features can be observed in Fig. 2.2.

2.2.1. Radial contribution

Here, we shall show that the radial integral IR(D) (2.24) has actually the analytical value of Eq.(2.27) in terms of the corresponding quantum numbers. Noting ∂

∂rR(r) ≡ R′(r), the integration by

parts leads us to the expression

IR(D) = 4∫ ∞

0rD−1 [R′(r)]2 dr = −4(D − 1)∫ ∞

0rD−2R(r)R′(r)dr − 4∫ ∞

0rD−1R(r)R′′(r)dr.

We make use now of the radial Schrodinger equation (2.12) that, in terms of the derivatives ofR(r), can be rewritten as

R′′(r) = a1(r)R′(r) + a2(r)R(r),with

a1(r) = 1 −Dr

(2.31)

a2(r) = l(l +D − 2)r2

− 2Zµ

r− 2Eµ. (2.32)


Thus, denoting R ≡ R(r) and a1,2 ≡ a1,2(r), the radial Fisher information yields

IR(D) = −4(D − 1)∫ ∞

0rD−2RR′dr − 4∫

∞

0rD−1R [a1R′ + a2R]dr

= −4∫ ∞

0rD−1 [(D − 1) + a1]RR′dr − 4∫ ∞

0rD−1a2R

2dr,

and the integration by parts in the first term and the use of the expectation value of a radialfunction given by Eq. (2.26) leads to

IR(D) = ∫ ∞

0[2(D − 1)(D − 2)

r2+ (D − 1)a1

r+ 2a′1 − 4a2]R2rD−1dr

= 2(D − 1)(D − 2) ⟨r−2⟩ + 2(D − 1) ⟨a1r⟩ + 2 ⟨a′1⟩ − 4 ⟨a2⟩ .

The explicit expression of a1 and a2, given by Eqs. (2.31) and (2.32), respectively, allows us tocalculate the desired expectation values in terms of the known ones of the form ⟨rα⟩ as

⟨a1r⟩ = (1 −D) ⟨r−2⟩

⟨a′1⟩ = (D − 1) ⟨r−2⟩⟨a2⟩ = l(l +D − 2) ⟨r−2⟩ − (2Zµ) ⟨r−1⟩ − 2Eµ.Inserting these values into the radial integral IR(D), we obtain

IR(D) = 4Zµ ⟨r−1⟩ − 4l(l +D − 2) ⟨r−2⟩ ,where, to eliminate the energy E we have made use of the Virial theorem

E = −Z2⟨r−1⟩ .

Thus, since ⟨r−1⟩ = Zµη2

and ⟨r−2⟩ has the value given in (2.29), it is straightforward to obtain the

expected expression (2.27) for the radial part of the D-dimensional position Fisher information ofa hydrogenic system.

2.2.2. Angular contribution

Let us now compute the angular integral IY(D) defined by Eq. (2.25). To do that, according toEq. (2.7) we express the involved hyperspherical harmonics in terms of the partial harmonics as

Yν(θ1, . . . , θD−2,0) = 1√2π

D−2

∏k=1

Y (k)νk,νk+1(θk).

Introducing this and the definition of the solid angle element (2.10) in the Eq. (2.25), we integratethe θD−1 angle and obtain, after some algebraic manipulation, that the integral IY(D) can bedecomposed into three types of integrals, i.e.,

IY(D) = D−2

∑i=1∫

π

0[ ∂∂θi

Y (i)νi,νi+1(θi)]2 (sin θi)D−i−1 dθi

×∏j<i∫

π

0[Y (j)νj ,νj+1

(θj)]2 (sin θj)D−j−3 dθj×∏

k>i∫

π

0[Y (k)νk,νk+1

(θk)]2 (sin θk)D−k−1 dθk.

2.3 The Fisher information in momentum space 15

Let us go through each of these terms one by one. The last set of integrals are exactly one dueto the normalization of the partial harmonics given in (2.8). The first integral can be solved byintegrating by parts, yielding

∫π

0[ ∂∂θi

Y (i)νi,νi+1(θi)]2 (sin θi)D−i−1 dθi = −∫ π

0

∂

∂θi[(sin θi)D−i−1 ∂

∂θiY (i)νi,νi+1

(θi)]Y (i)νi,νi+1(θi)dθi

= νi(νi +D − i − 1) − νi+1(νi+1 +D − i − 2)∫ π

0(sin θi)D−i−3 [Y (i)νi,νi+1

(θi)]2 dθi,where we have used the differential equation (2.6) for which the partial harmonics are solutions.Thus, to calculate the result of the angular part of the Fisher information IY(D) we only have toevaluate the integral

∫π

0[Y (j)νj ,νj+1

(θj)]2 (sin θj)D−j−3 dθj = 2νj +D − j − 12νj+1 +D − j − 2 . (2.33)

This task is performed by writing the partial harmonics in terms of the associated Legendre func-tions Pm

n (z) [55] and using the expression

∫1

−1

1

1 − z2 [Pmn (z)]2 dz = (n +m)!

m(n −m)! . (2.34)

The final result yields

IY(D) = D−2

∑i=1

(νi(νi +D − i − 1) − νi+1(νi+1 +D − i − 2)(2νi +D − i − 1)2νi+1 +D − i − 2 )∏

j<i

2νj +D − j − 12νj+1 +D − j − 2

= (2ν1 +D − 2)D−2∑i=1

(νi(νi +D − i − 1)2νi +D − i − 1 − νi+1(νi+1 +D − i − 2)

2νi+1 +D − i − 2 )= ν1(ν1 +D − 2) − (2ν1 +D − 2)∣νD−1∣

2,

that can be expressed as (2.28) substituting ν1 ≡ l = L − (D − 3)/2.2.3. The Fisher information in momentum space

Here we calculate the Fisher information of the D-dimensional hydrogenic system in momentumspace in a parallel way to the position space described in the previous section. This measure isdefined as

Ip(D) = ∫RD

[∇Dγ (p)]2γ (p) dp, (2.35)

where γ(p) = ∣Φη,L,ν(p)∣2 and dp = pD−1dpdΩpD−1

are the hydrogenic probability distributiongiven in (2.20) and volume element of the D-dimensional momentum space, respectively. Here,the gradient operator has the same formal expression as the one given in (2.3) but changing r byp. Since the angular dependence is again given by the hyperspherical harmonics, once more theprobability distribution is independent of the coordinate θp

D−1, i.e.,

γ(p) = ∣Φη,L,ν(p, θp1 , . . . , θpD−1)∣2 = [Φη,L,ν(p, θp1 , . . . , θpD−2,0)]2 ,so that one can rewrite the Fisher Information in momentum space as

Ip(D) = 4∫RD[∇DΦη,L,ν(p, θ1, . . . , θD−2,0)]2 dp.


010

2010

20

300

2

4

6

8x 10

6

Lη

a

0 10 20 300

2

4

6

8x 10

6

η

I r

b

10 20 300

5

10x 106

L

I r

c

Figure 2.3.: a: Fisher information in momentum space of the D-dimensional hydrogen atom (Z = 1) varyingthe grand principal and orbital quantum numbers η and L, respectively, for νD−1 = 0. Note that, for eachvalue of η, we have L = 0, . . . , η − 1. b: For different values of L and νD−1 = 0, variation of the Fisherinformation with η. c: Again for νD−1 = 0, the dependence of the Fisher information in momentum spacewith the generalized orbital quantum number for fixed values of η.

Exploiting the separation of variables exposed in equation (2.19) allows us to write

Ip(D) = 4JP (D) + 4 ⟨p−2⟩ IY(D).where we denote

JP (D) = ∫ ∞

0[ ∂∂pM(p)]2 pD−1dp, (2.36)

whose solution will be found in the next section, and where the expectation value ⟨p−2⟩ has thewell-known expression

⟨p−2⟩ = ∫ ∞

0p−2M2(p)pD−1dp = ( η

Zµ)2 8η − 3(2L + 1)

2L + 1 , (2.37)

where we have used the notation M(p) ≡Mη,L(p). The angular integral IY(D) has been shown tobe reproduced by (2.28) in the previous section. Gathering together the radial integral JP (D), theexpectation value ⟨p−2⟩ and the angular integral IY(D), one finally obtains the following expressionfor the Fisher information of a D-dimensional hydrogenic system in momentum space:

Ip(D) = 2( η

Zµ)2 [5η2 − 3L(L + 1) − ∣νD−1∣(8η − 3(2L + 1)) + 1] , (2.38)

valid for D ≥ 2. In the Fig. 2.3, the case of the hydrogen (Z = 1) is represented. From (2.38) onecan easily see that the Fisher information in momentum space has again a linear dependence withthe absolute value of grand magnetic quantum number ∣νD−1∣. Thus, we take the case νD−1 = 0in Fig. 2.3 and study the more complicated dependence on the quantum numbers η and L. It isevident that the Fisher information in momentum space decreases as L increases, increasing on theother hand with the generalized principal quantum number. Thus, opposite to the case of positionspace, the extreme value of this measure is not obtained for the ground state of the atom (minimumpossible value of η), but for the highly excited states with low orbital quantum numbers.

2.3 The Fisher information in momentum space 17

2.3.1. Radial contribution

Let us now evaluate the radial integral JP (D) given by Eq. (2.36). Here we shall prove that thisintegral has the value

JP (D) = ( ηZµ)2 [(2L −D + 3)(2L +D − 1)(6L − 8η + 3)

2L + 1 + 2(1 + 5η2 − 3L(L + 1))] . (2.39)

To do so, we start by making the change of variable x = ηZµp so that

JP (D) = (Zµη)D−2∫ ∞

0[ ∂∂xM(x)]2 xD−1dx,

with

Mη,L(x) = Kη,LxL−

D−32(1 + x2)L+2CL+1

η−L−1 (1 − x21 + x2) .A further change of variable x → y ∶ y = x2−1

x2+1, so that dx = (1 + y)−1/2(1 − y)−3/2dy, allows us to

write

JP (D) = (Zµη)D−2∫ +1

−1[ ∂∂yM(y)]2 (1 + y)D/2(1 − y)−D/2+2dy,

with

Mη,L(y) = Kη,L(−1)η−L−12L+2

(1 + y)L2 −D−34 (1 − y)L2 +D+5

4 CL+1η−L−1(y),

where the parity of the Gegenbauer polynomials Cλm(−y) = (−1)mCλ

m(y) has been used. With thenotation ωλ(y) = (1 − y2)λ−1/2 for the weight function of the Gegenbauer polynomials Cλ

m(y) andafter some algebraic manipulation, JP (D) can be decomposed into six non-zero integrals,

JP (D) = (Zµη)D−2 K2

η,L

22L+4[JP1

+ JP2+ JP3

+ JP4+ JP5

+ JP6] .

The explicit expression of these JPione has to evaluate is

JP1= (2L −D + 3)2 ∫ 1

−1ωL+1(y) [CL+1

η−L−1(y)]2 (1 + y)−1dyJP2

= [(D + 1)24

− (2L −D + 3)2]∫ 1

−1ωL+1(y) [CL+1

η−L−1(y)]2 dyJP3

= [L(D − 3L − 11) + 2D − 10]∫ 1

−1ωL+1(y) [CL+1

η−L−1(y)]2 y2dyJP4

= 4(L + 1)2 ∫ 1

−1ωL+2(y) [CL+2

η−L−2(y)]2 (1 + 3y2)dyJP5

= −8(2L −D + 3)(L + 1)∫ 1

−1ωL+1(y)CL+1

η−L−1(y)CL+2η−L−2(y)ydy

JP6= 2(6L −D + 11)(L + 1)∫ 1

−1ωL+2(y)CL+1

η−L−1(y)CL+2η−L−2(y)ydy,


and their values are

JP1= Aη,L

2η(2L −D + 3)22L + 1

JP2= Aη,L [(D + 1)2

4− (2L −D + 3)2]

JP3= Aη,L [L(D − 3L − 11) + 2D − 10] η2 − 1 −L(L + 1)

2(η2 − 1)JP4

= Aη,L [η2 − (L + 1)2] [1 + 3

2

η2 − 3 −L(L + 3)η2 − 1 ]

JP5= −Aη,L4(2L −D + 3)(η −L − 1)

JP6= Aη,L(6L −D + 11)(L + 1)η2 − (L + 1)2

2(η2 − 1) ,

with

Aη,L = π(η +L)!22L+1η(η −L − 1)!(L!)2 .

All the integrals appearing have be calculated by making intensive use of the recurrence and or-thogonality relations verified by the Gegenbauer polynomials, i.e.,

yCλn(y) = Cλ

n+1(y) − 2λ + n − 12λ − 2 Cλ−1

n+1(y),2(λ + n)yCλ

n(y) = (n + 1)Cλn+1(y) + (2λ + n − 1)Cλ

n−1(y)and

∫1

−1ωλ(y)Cλ

m(y)Cλn(y)dy = π21−2λΓ(n + 2λ)n!(n + λ)Γ(λ)2 δmn,

respectively. Note that, in particular for the first integral JP1, the expression of the Gegenbauer

polynomials in terms of the associated Legendre functions [55] and the relation (2.34) are needed.It can be seen that, eventually, the result (2.39) is obtained.

2.4. Alternative method and application to the case D=3

In another work [56], an alternative way of finding the position and momentum Fisher informationin terms of the radial expectation values ⟨rk⟩ and ⟨pk⟩, with k = −2,2 has been proposed. Moreover,an uncertainty Fisher information relation is obtained for multidimensional single-particle systemswith general central potentials. We outline here the alternative derivation of the expressions (2.30)and (2.38) for the Fisher informations and we apply them to the real (D = 3) hydrogenic atoms.

Let us start with the position space. The momentum expectation value ⟨p2⟩ = ⟨∇2D⟩ is given by

⟨p2⟩ = ∫RD∣∇DΨη,L,ν(r, θ1, . . . , θD−1)∣2 dr. (2.40)

On the other hand, as we stated in Section 2.2, given that in a central potential the angular partof the wavefunction is described by the hyperspherical harmonics (2.7), and due to the dependenceof these functions with the angular coordinate θD−1, the probability distribution in position spacecan be written as in Equation (2.22). As a consequence, the Fisher information can be expressed as(2.23). According to the form of the multidimensional gradient (2.3), the expectation value (2.40)can be decomposed as

⟨p2⟩ = ∫RD∣∇DΨη,L,ν(r, θ1, . . . , θD−2,0)∣2 dr +K(D),

2.5 Summary and conclusions 19

with

K(D) = ∫RD∣1r

D−2

∏k=1

(sin θk)−1 ∂

∂θD−1Ψη,L,ν(r, θ1, . . . θD−1)∣

2

dr

= ⟨r−2⟩D−2∏k=1∫

π

0∣Y (k)νk,νk+1

(θk)∣2 (sin θk)D−k−3 dθk 1

2π∫

2π

0∣ ∂

∂θD−1eiνD−1θD−1 ∣2 dθD−1

= ⟨r−2⟩ ∣νD−1∣2(2L + 1),

where the radial expectation value definition (2.26) and the integral (2.33) have been used. As aresult, the Fisher information can be expressed as

Ir(D) = 4 ⟨p2⟩ − 2K(D).The combination of these results leads to the following relation between the Fisher information andthe radial expectation values ⟨r−2⟩ and ⟨p2⟩,

Ir(D) = 4 ⟨p2⟩ − 2 ∣νD−1∣ (2L + 1) ⟨r−2⟩ . (2.41)

The application of the same procedure in momentum space allows to obtain that

Ip(D) = 4 ⟨r2⟩ − 2 ∣νD−1∣ (2L + 1) ⟨p−2⟩ . (2.42)

When applied to the hydrogenic case, these expressions give the same results as the ones obtainedin the previous sections by direct integration.

Finally, it is worth noting that these results, when D = 3 is taken, coincide with the valuesobtained by other means [57, 58], i.e.,

Ir(D = 3) = 4Z2µ2

n3(n − ∣m∣)

Ip(D = 3) = 2n2

Z2µ2[5n2 + 1 − 3l(l + 1) − ∣m∣ (8n − 6l − 3)] .

One observes that the Fisher information of the hydrogenic system in position space does not dependon the orbital quantum number l (as in the momentum space case), but only on the principal andmagnetic quantum numbers n and m, respectively.

2.5. Summary and conclusions

The translationally invariant Fisher information, which is an information-theoretic measure of thelocalization of the quantum-mechanical distribution density all over the space, has been determinedin a closed and compact form for D-dimensional hydrogenic systems. This has been done for bothposition and momentum spaces, and the corresponding expressions have been found to depend onthe dimension parameter D, the nuclear charge Z and the D quantum numbers which fully describethe physical state under consideration.

Most of the results presented in this chapter are published in Refs. [57, 59].

3. Parameter-based Fisher Information of

orthogonal polynomials

In 2005, J.S. Dehesa and J. Sanchez-Ruiz [31] exactly derived the locality Fisher information of alarge class of probability distributions, the Rakhmanov densities, defined by

ρn(x) = 1

d2np2n(x)ω(x)χ[a,b](x),

where χ[a,b](x) is the characteristic function for the interval [a, b], and pn(x) denotes a sequenceof real polynomials orthogonal with respect to the nonnegative definite weight function ω(x) onthe interval [a, b] ⊆ R, that is,

∫b

apn(x)pm(x)ω(x)dx = d2nδn,m (3.1)

with deg pn(x) = n. As first pointed out by E.A. Rakhmanov [60] (see also [61]), these distributionsplay a fundamental role in the analytic theory of orthogonal polynomials. In particular, it has beenshown that they govern the asymptotic behavior of the ratio pn+1(x)/pn(x) as n→∞. On the otherhand, the Rakhmanov densities of the classical orthogonal polynomials of a real continuous variabledescribe the quantum-mechanical probability distributions of ground and excited states of numerousphysical systems with an exactly solvable Schrodinger equation, particularly the most commonprototypes (harmonic oscillator, hydrogen atom,. . . ), in position and momentum spaces. These twofundamental and applied reasons have motivated an increasing interest for the determination of thespreading of the classical orthogonal polynomials pn(x) throughout its interval of orthogonalityby means of the information-theoretic measures of their corresponding Rakhmanov densities ρn(x)[16, 29, 31, 32, 57, 59, 62, 63].

The Shannon entropy of these densities has already been examined numerically [32]. On the the-oretical side, let us point out that its asymptotics (n →∞) is well known for all classical orthogonalpolynomials, but its exact value for every fixed n is only known for Chebyshev polynomials [19] andsome Gegenbauer polynomials [30]. To this respect, see Ref. [29], which reviews the knowledge upto 2001. The variance and Fisher information of the Rakhmanov densities have also been found ina closed and compact form for all classical orthogonal polynomials [16, 31]. For other functionalsof these Rakhmanov densities, see Ref. [28].

In this chapter we shall calculate the Fisher information of the real and continuous classicalorthogonal polynomials (Gegenbauer, Grosjean, Jacobi and Laguerre) with respect to the param-eter(s) of the polynomials. We start collecting some basic properties of these classical orthogonalpolynomials which will be used later on. Then, the Fisher information with respect to a param-eter is fully determined for Jacobi and Laguerre polynomials first, and then for Gegenbauer andGrosjean polynomials. Finally, conclusions and some open problems are given.

22 Parameter-based Fisher Information of orthogonal polynomials

3.1. Some properties of the parameter-dependent classical orthogonal

polynomials

Let yn(x; θ)n∈N0stand for the sequence of polynomials orthonormal with respect to the nonneg-

ative definite weight function ω(x; θ) on the real support (a, b), so that

∫b

ayn(x; θ)ym(x; θ)ω(x; θ)dx = δnm, (3.2)

with deg yn = n. Here, in particular, we shall consider the classical families of Laguerre L(α)n (x),

α > −1, and Jacobi J(α,β)n (x), α,β > −1, polynomials. The normalized-to-unity density functions

ρn(x; θ), defined as

ρn(x; θ) = y2n(x; θ)ω(x; θ) (3.3)

are called Rakhmanov densities [60, 61]. Here, we gather various properties of the parameter-dependent classical orthogonal polynomials in a real and continuous variable (i.e., Laguerre andJacobi) in the form of two lemmas, which shall be used later on.

The weight function of these polynomials can be always written as

ω(x; θ) = h(x) [t(x)]θ .In particular,

ωL(x;α) = e−xxα Ô⇒ hL(x) = e−x and tL(x) = xfor the Laguerre polynomials, L

(α)n (x), and

ωJ(x;α) = (1 − x)α(1 + x)β Ô⇒ hJ(x) = (1 + x)β and tJ(x) = 1 − xfor the Jacobi, P

(α,β)n (x). Note that in this definition we consider the parameter of interest to be

the first one, i.e., α. When considering the second one, we take

ωJ(x;β) = (1 − x)α(1 + x)β Ô⇒ hJ(x) = (1 − x)α and tJ(x) = 1 + x.Lemma 3.1.1 The derivative of the orthonormal polynomial yn(x; θ) with respect to the parameterθ is given by

∂

∂θyn(x; θ) = n

∑k=0

Ak(θ)yk(x; θ) (3.4)

with

Ak(θ) = dk(θ)dn(θ)Ak(θ) for k = 0,1, . . . , n − 1 (3.5)

An(θ) = An(θ) − 1

dn(θ)∂

∂θdn(θ), (3.6)

where d2m(θ) denotes, according to Eq. (3.1), the normalization constant of the orthogonal polyno-mial pm(x) = ym(x; θ), and Ak(θ) with k = 0,1, . . . ,m are the expansion coefficients of the derivativeof ym(x; θ) in terms of the system ym(x; θ), i.e.,

∂

∂θym(x; θ) = m

∑k=0

Ak(θ)yk(x; θ). (3.7)

3.1 Some properties of the parameter-dependent classical orthogonal polynomials 23

Both quantities dn(θ) and Ak(θ) are known in the literature for the Laguerre and Jacobi cases.

Indeed, the norms for the Laguerre L(α)n (x) and the Jacobi P

(α,β)n (x) polynomials [64] are

[d(L)n (α)]2 = Γ(n +α + 1)n!

, (3.8)

[d(J)n (α,β)]2 = 2α+β+1Γ(n +α + 1)Γ(n + β + 1)n!(2n + α + β + 1)Γ(n + α + β + 1) , (3.9)

respectively. On the other hand, the expansion coefficients in Eq. (3.7) are known to have the form[65, 66, 67]

A(L)k= 1

n − k for k = 0,1, . . . , n − 1 and A(L)n = 0 (3.10)

for the Laguerre polynomials L(α)n (x) and

A(Jα)k

= α + β + 1 + 2k(n − k)(α + β + 1 + n + k) (β + k + 1)n−k(α + β + k + 1)n−k ; k = 0,1, . . . , n − 1, (3.11)

A(Jα)n = n−1

∑k=0

1

α + β + 1 + n + k = ψ(1 +α + β + 2n) − ψ(1 + α + β + n) (3.12)

for the Jacobi expansion of ∂∂αP(α,β)n (x), and

A(Jβ)k

= (−1)n−k α + β + 1 + 2k(n − k)(α + β + 1 + n + k) (α + k + 1)n−k(α + β + k + 1)n−k ; k = 0,1, . . . , n − 1, (3.13)

A(Jβ)n = n−1

∑k=0

1

α + β + 1 + n + k = ψ(1 + α + β + 2n) −ψ(1 +α + β + n) (3.14)

for ∂∂βP(α,β)n (x). The symbol ψ(x) = Γ′(x)

Γ(x) denotes here the well-known digamma function.

Using Eqs. (3.8) and (3.10) in Lemma 3.1.1 allows us to derive the following expression for the

expansion coefficients Ak(α) for the orthonormal Laguerre polynomials L(α)n (x)

A(L)k

= 1

n − k [ (k + 1)n−k(k +α + 1)n−k ]1/2

; k = 0,1, . . . , n − 1 (3.15)

A(L)n = −ψ(n + α + 1)2

. (3.16)

In a similar way, this Lemma, together with the specific values for the Jacobi polynomials (3.9)and (3.12)-(3.14), has allowed us to find the expressions

A(Jα)k

= [ (k + β + 1)n−k(k + 1)n−k(k +α + 1)n−k(k + α + β + 1)n−k2n +α + β + 12k +α + β + 1]

1/2(3.17)

× 2k + α + β + 1(n − k)(n + k + α + β + 1) ; k = 0,1, . . . , n − 1A(Jα)n = 1

2[2ψ(2n + α + β + 1) −ψ(n +α + β + 1) −ψ(n +α + 1)− ln 2 + 1

2n + α + β + 1] (3.18)


and

A(Jβ)k

= [ (k + α + 1)n−k(k + 1)n−k(k + β + 1)n−k(k +α + β + 1)n−k2n + α + β + 12k + α + β + 1]

1/2(3.19)

×(−1)n−k 2k +α + β + 1(n − k)(n + k + α + β + 1) ; k = 0,1, . . . , n − 1A(Jβ)n = 1

2[2ψ(2n + α + β + 1) − ψ(n + α + β + 1) − ψ(n + β + 1)− ln 2 + 1

2n +α + β + 1] (3.20)

for the expansion coefficients of the derivative of the Jacobi polynomials P(α,β)n (x) with respect to

the parameters α and β, respectively.

Lemma 3.1.2 The parameter-dependent classical orthonormal polynomials yn(x; θ) satisfy(a) ∫

b

a

∂ω(x; θ)∂θ

[yn(x; θ)]2 dx = −2An(θ)(b) ∫

b

a

∂ω(x; θ)∂θ

yn(x; θ)yk(x; θ)dx = −Ak(θ) k = 0,1, . . . , n − 1(c) ∫

b

a

∂2ω(x; θ)∂θ2

[yn(x; θ)]2 dx = 2 n

∑k=0

(Ak(θ))2 + 2(An(θ))2 − 2∂An(θ)∂θ

.

To prove the integrals (a) and (b) we have to derive with respect to the parameter θ the orthonor-malization condition (3.2) for m = n and m = k ≠ n, respectively. Then, one has to use the Lemma3.1.1 and again Eq. (3.2), and the results follow. The integral (c) is obtained by deriving theintegral (a) with respect to θ and taking into account the values of the two previous, i.e., (a) and(b).

3.2. Parameter-based Fisher information of Jacobi and Laguerre

polynomials

The distribution of the orthonormal polynomials yn(x; θ) on their orthonormality interval and thespreading of the associated Rakhmanov densities can be most appropriately estimated by meansof their information-theoretic measures, the Shannon entropy [39] and the Fisher information [33].The former has been theoretically [29, 62] and numerically [32] examined for general orthogonalpolynomials. In the latter case, the Fisher information associated with translations of the variable(i.e., the locality Fisher information) has been analyzed both analytically [31] and numerically[63]. Here we extend this study by means of the computation of a more general concept, theparameter-based Fisher information of the polynomials yn(x; θ). This quantity is defined as theFisher information of the associated Rakhmanov density (3.3) with respect to the parameter θ, i.e.,according to Eq. (1.1) and after some manipulation,

In(θ) = 4∫ b

a ∂∂θ[ρn(x; θ)]1/22 dx. (3.21)

Theorem 3.2.1 The parameter-based Fisher information In(θ) of the parameter-dependent clas-sical orthonormal polynomials yn(x; θ) (i.e., Jacobi and Laguerre) defined by Eq. (3.21) has thevalue

In(θ) = 2n−1

∑k=0

[Ak(θ)]2 − 2∂An(θ)∂θ

, (3.22)

3.2 Parameter-based Fisher information of Jacobi and Laguerre polynomials 25

where Ak(θ), k = 0,1, . . . , n are the expansion coefficients of the derivative with respect to θ ofyn(x; θ) in terms of the polynomials yk(x; θ)nk=0, which are given by Lemma 3.1.1. See Eqs.(3.15)-(3.16) and (3.17)-(3.20) for the Laguerre and Jacobi families, respectively.

Remark Let us underline that the Fisher information with respect to a parameter in the cases oforthogonal, monic orthogonal and orthonormal polynomials has the same value. This is because ofthe definition of the Rakhmanov densities (3.3) and their probabilistic character.

Proof To prove this theorem we start taking the derivative with respect to θ of [ρn(x; θ)]1/2 andusing Lemma 3.1.1 to obtain

∂

∂θ[ρn(x; θ)]1/2 = [ω(x; θ)]1/2 n

∑k=0

Ak(θ)yk(x; θ) + ∂ [ω(x; θ)]1/2∂θ

yn(x; θ).Introducing this expression into Eq. (3.21) leads to

In(θ) = J1 + J2 + J3,where

J1 = 4∫ b

aω(x; θ)( n

∑k=0

Ak(θ)yk(x; θ))2 dx,J2 = 4∫ b

a

⎛⎝∂ [ω(x; θ)]1/2

∂θ

⎞⎠2 [yn(x; θ)]2 dx,

and

J3 = 8 n

∑k=0

Ak(θ)∫ b

a[ω(x; θ)]1/2 ∂ [ω(x; θ)]1/2

∂θyn(x; θ)yk(x; θ)dx.

Now we take into account that the weight function of the parameter-dependent families of classicalorthonormal polynomials in a real and continuous variable (i.e., Laguerre and Jacobi) has the formω(x; θ) = h(x) [t(x)]θ, so that

⎧⎪⎪⎨⎪⎪⎩∂ [ω(x; θ)]1/2

∂θ

⎫⎪⎪⎬⎪⎪⎭2

= 1

4ω(x; θ) [ln t(x)]2 = 1

4

∂2ω(x; θ)∂θ2

,

and

[ω(x; θ)]1/2 ∂ [ω(x; θ)]1/2∂θ

= 1

2ω(x; θ) ln t(x) = 1

2

∂ω(x; θ)∂θ

.

The use of these two expressions in the integrals J2 and J3 together with Lemma 3.1.2 and theconsideration of the orthonormalization condition (3.2) in J1 yield

J1 = 4 n

∑k=0

[Ak(θ)]2 ,J2 = 2 n

∑k=0

[Ak(θ)]2 + 2 [An(θ)]2 − 2∂An(θ)∂θ

,

and

J3 = −4 n

∑k=0

[Ak(θ)]2 − 4 [An(θ)]2 ,so that they lead to Eq. (3.22).


Corollary 3.2.1.1 The Fisher information with respect to the parameter α, I(L)n (α), of the La-

guerre polynomial L(α)n (x) is given by

I(L)n (α) = 2n−1

∑k=0

[A(L)k]2 − 2∂A(L)n

∂α(3.23)

= ψ(1)(n +α + 1) + 2n

n + α4F3

⎛⎜⎝1 1 1 1 − n

12 2 1 − α − n

⎞⎟⎠ ,where ψ(1)(x) = d

dxψ(x) is the trigamma function.

Corollary 3.2.1.2 The Fisher information with respect to the parameter α, I(Jα)n (α,β), of the

Jacobi polynomial P(α,β)n (x) is given by

I(Jα)n (α,β) = 2n−1

∑k=0

[A(Jα)k]2 − 2∂A(Jα)n

∂α

= 2Γ(n + β + 1)n!(2n + α + β + 1)Γ(n +α + 1)Γ(n +α + β + 1)× n−1

∑k=0

Γ(k + α + 1)Γ(k + α + β + 1)(2k + α + β + 1)Γ(k + β + 1)k!(n − k)2(n + k + α + β + 1)2

−2ψ(1)(2n +α + β + 1) +ψ(1)(n +α + β + 1)+ψ(1)(n +α + 1) + 1(2n +α + β + 1)2 , (3.24)

and the Fisher information with respect to the parameter β, I(Jβ)n (α,β), of the Jacobi polynomial

P(α,β)n (x) is given by

I(Jβ)n (α,β) = 2

n−1

∑k=0

[A(Jβ)k]2 − 2∂A(Jβ)n

∂β

= 2Γ(n + α + 1)n!(2n + α + β + 1)Γ(n + β + 1)Γ(n +α + β + 1)

× n−1

∑k=0

Γ(k + β + 1)Γ(k +α + β + 1)(2k + α + β + 1)Γ(k +α + 1)k!(n − k)2(n + k + α + β + 1)2

−2ψ(1)(2n + α + β + 1) + ψ(1)(n +α + β + 1)+ψ(1)(n + β + 1) + 1(2n + α + β + 1)2 . (3.25)

Both corollaries follow from Theorem 3.2.1 in a straightforward manner by taking into account theexpressions (3.15)-(3.20) for the expansion coefficients Ak of the corresponding families.

3.3. Parameter-based Fisher information of the Gegenbauer and

Grosjean polynomials

In this section we describe the Fisher information of two important subfamilies of the Jacobi

polynomials P(α,β)n (x): the ultraspherical or Gegenbauer polynomials [29, 64, 68], which have

α = β = λ − 1/2, and the Grosjean polynomials of the first and second kind [69, 70, 71], whichhave α + β = ±1, respectively. Let us remark that the parameter-based Fisher information for

3.3 Fisher information of Gegenbauer and Grosjean polynomials 27

these subfamilies cannot be obtained from the expressions of the similar quantity for the Jacobipolynomials (given by corollary 3.2.1.2) by means of a mere substitution of the parameters, becauseit depends on the derivative with respect to the parameter(s) and now α and β are correlated.

The Gegenbauer polynomials C(λ)n (x) are Jacobi-like polynomials satisfying the orthogonality

condition (3.1) with the weight function

ωC(x;λ) = (1 − x2)λ− 12 for λ > −1

2,

and the normalization constant

[d(C)n (λ)]2 = π21−2λΓ(n + 2λ)n!(n + λ)Γ2(λ)

so that they can be expressed as

C(λ)n (x) = (2λ)n(λ + 12)nP

(λ− 12,λ− 1

2)

n (x).It is known [66] that the expansion (3.7) for the derivative of C

(λ)n (x) with respect to the parameter

λ has the coefficients

A(C)k(λ) = 2(1 + (−1)n−k) (k + λ)(k + n + 2λ)(n − k) for k = 0,1, . . . , n − 1

A(C)n (λ) = n−1

∑k=0

2(k + 1)(2k + 2λ + 1)(k + 2λ) + 2

k + n + 2λ = ψ(n + λ) −ψ(λ).Then, according to Lemma 3.1.1, the expansion (3.4) for the derivative of the orthonormal Gegen-bauer polynomials has the following coefficients

A(C)k(λ) = [Γ(k + 2λ)n!(n + λ)

Γ(n + 2λ)k!(k + λ)]1/2 2(1 + (−1)n−k) (k + λ)(k + n + 2λ)(n − k) for k = 0,1, . . . , n − 1

A(C)n (λ) = ψ(n + λ) − ψ(n + 2λ) + ln 2 + 1

2(n + λ) .Theorem 3.2.1 provides, according to Eq. (3.22), the following value for the Fisher information of

the Gegenbauer polynomials C(λ)n (x) with respect to the parameter λ:

I(C)n (λ) = 16n!(n + λ)Γ(n + 2λ)

n−1

∑k=0

(1 + (−1)n−k)Γ(k + 2λ)(k + λ)k!(k + n + 2λ)2(n − k)2

−2ψ(1)(n + λ) + 4ψ(1)(n + 2λ) + 1(n + λ)2 .Let us now perform the same task for the Grosjean polynomials of the first and second kind,

which are the monic Jacobi polynomials P(α,β)n (x) with α+β = −1,+1, respectively. Hence, we have

[69, 70]G(α)n (x) = cnP (α,−1−α)n (x) − 1 < α < 0,

andg(α)n (x) = enP (α,1−α)n (x) − 1 < α < 2,

for the Grosjean polynomials of first and second kind, respectively, with the values

cn = 2n(2n − 1n)−1 and en = 2n(2n + 1

n)−1.


The Grosjean polynomials of the first kind G(α)n (x) satisfy the orthogonality condition (3.1) with

respect to the weight function

ωG(x;α) = (1 − x1 + x)

α 1

1 + x,and with normalization constant

[d(G)n (α)]2 = 22n−1Γ2(n)Γ2(2n) Γ(n + α + 1)Γ(n − α).

These polynomials, together with the Chebyshev of the first, second, third and fourth kind, arethe only Jacobi polynomials for which the associated ones are again Jacobi polynomials [69]. The

expansion (3.7) for the derivative of G(α)n (x) with respect to the parameter α can be obtained as

∂G(α)n (x)∂α

= ∂P (α,−1−α)n (x)∂α

= ∂P (α,β)n (x)∂α

RRRRRRRRRRRβ=−1−α −∂P(α,β)n (x)∂β

RRRRRRRRRRRβ=−1−α =n−1

∑k=0

A(G)k(α)G(α)n (x)

with

A(G)k(α) = 2n−k+1k

n2 − k2Γ(2k)Γ(n + 1)Γ(2n)Γ(k + 1) [(k − α)n−k − (−1)n−k(k +α + 1)n−k]

for k = 0,1, . . . , n − 1 and A(G)n (α) = 0. Then, Lemma 3.1.1 provides the expansion (3.4) for the

derivative of the orthonormal Grosjean polynomials with the coefficients

A(G)k(α) = 2n

n2 − k2(k − α)n−k − (−1)n−k(k + α + 1)n−k[(k +α + 1)n−k(k − α)n−k]1/2 for k = 0,1, . . . , n − 1,

A(G)n (α) = 1

2[ψ(n − α) −ψ(n +α + 1)] .

Finally, Eq. (3.22) of Theorem 3.2.1 allows us to find the following value for the Fisher informationof the Grosjean polynomials of the first kind:

I(G)n (α) = 8n2n−1

∑k=0

1(n2 − k2)2[(k − α)n−k − (−1)n−k(k + α + 1)n−k]2(k +α + 1)n−k(k − α)n−k

+ψ(1)(n −α) + ψ(1)(n + α + 1).On the other hand, the Grosjean polynomials of the second kind g

(α)n (x) satisfy the orthogonality

property (3.1) with respect to the weight function

ωg(x;α) = (1 − x1 + x)

α (1 + x),and the normalization constant

[d(g)n (α)]2 = 22n+1Γ2(n)Γ2(2n + 2) Γ(n +α + 1)Γ(n − α + 2).

Moreover, the derivative of these polynomials with respect to the parameter α can be again ex-panded in the form (3.4) as

∂g(α)n (x)∂α

= ∂P (α,1−α)n (x)∂α

= ∂P (α,β)n (x)∂α

RRRRRRRRRRRβ=1−α −∂P(α,β)n (x)∂β

RRRRRRRRRRRβ=1−α =n−1

∑k=0

A(g)k(α)g(α)n (x),


with

A(g)k(α) = 2n−k+1(k + 1)(n − k)(n + k + 2) Γ(2k + 2)Γ(n + 1)Γ(2n + 2)Γ(k + 1) [(k + 2 − α)n−k − (−1)n−k(k +α + 1)n−k]

for k = 0,1, . . . , n−1 and A(g)n (α) = 0. Then, Lemma 3.1.1 is able to provide the analogous expansion

(3.4) for the orthonormal polynomials with the coefficients

A(g)k(α) = 2(k + 1)(n − k)(n + k + 2) (k + 2 − α)n−k − (−1)

n−k(k +α + 1)n−k[(k +α + 1)n−k(k + 2 − α)n−k]1/2 ;

for k = 0,1, . . . , n − 1,A(g)n (α) = 1

2[ψ(n + 2 − α) −ψ(n +α + 1)] .

Finally, we obtain by means of Eq. (3.22) of Theorem 3.2.1 the Fisher information of the Grosjeanpolynomials of the second kind, which turns out to have the value

I(g)n (α) = 8n−1

∑k=0

(k + 1)2(n − k)2(n + k + 2)2 [(k + 2 − α)n−k − (−1)n−k(k +α + 1)n−k]2(k + α + 1)n−k(k + 2 −α)n−k

+ψ(1)(n + 2 −α) + ψ(1)(n + α + 1).3.4. Summary and conclusions

In summary, we have calculated the parameter-based Fisher information for the classical orthogonalpolynomials of a continuous and real variable with a parameter dependence; namely, the Jacobi andLaguerre polynomials. Then we have evaluated the corresponding Fisher information for the two

most relevant parameter-dependent Jacobi polynomials P(α,β)n (x): the Gegenbauer (α = β = λ−1/2)

and the Grosjean (α + β = ±1) families.As we already pointed out, the Fisher information is a measure of the error when estimating the

considered parameter. In practice, this can be understood as follows: Let us consider a probabilitydistribution (here the Rakhmanov density) with two different but close values of the parameter. Ifthose distributions are very similar the error associated to a measure of this parameter will be verylarge and, as a consequence of the Cramer-Rao inequality (1.2), the value of the Fisher informationwill be small. Thus, we can consider the Fisher information in this sense to be a measure of thedistinguishability of a probability distribution with respect to the neighboring ones when varyinga given parameter. This can also be seen from the relation between the Fisher information andthe so-called Kullback-Leibler divergence (1.5). This relative entropy between two distributions,that was given in Eq. (1.6), measures the difference in information when using a distribution q(y)different from the reference one p(y).

This outlined use of the Fisher information as a measure of distinguishability of the probabilitydistributions can be seen, e.g., in Fig. 3.1 for the Rakhmanov densities associated to the Laguerrepolynomials. The Fisher information in this case increases with the degree n and decreases withthe parameter α. In Figs. 3.1b and c, two limiting cases are depicted, with large value of n andsmall of α and viceversa. One can see that, as expected, the distributions are in the first casequalitatively different, corresponding to a large value of the Fisher information, while in the lattercase the opposite behavior is found. Quantitatively, the Kullback-Leibler divergence is found tobe D [ρ9(x; 0.1)∥ρ9(x; 1.1)] = 0.95 and D [ρ0(x; 9.5)∥ρ0(x; 10.5)] = 0.048, respectively. The samefeatures are observed in Fig. 3.2 for the Rakhmanov densities of the Gegenbauer polynomials. Here,there is again a decreasing value of the Fisher information for increasing λ, while it decreases withthe degree of the polynomial n (except for small values of λ, where there is a sudden increase followed


0

5

10 0

5

100

1

2

3

nα

a

0 1 20

0.2

0.4

0.6

x

ρ 9(x;α

)

b

0 10 200

0.05

0.1

xρ 0(x

;α)

c

α=0.1α=1.1

α=9.5α=10.5

Figure 3.1.: a: Fisher information of the Rakhmanov density associated to the orthonormal Laguerre poly-

nomials L(α)n (x); dependence with n and α. b: Distributions for n = 9 and α = 0.1,1.1, where the Fisher

information has the largest value. One observes that the two distributions are qualitatively different. c:

Conversely, the Fisher information has a very low value for n = 0 and α = 9.5,10.5, and the distributions canbe seen to be, in fact, qualitatively similar.

0

5

10 0

5

100

1

2

3

4

nλ

a

−1 0 10

1

2

x

ρ 2(x;λ

)

b

−1 0 10

1

2

x

ρ 0(x;λ

)

c

λ=0.1λ=1.1

λ=9.5λ=10.5

Figure 3.2.: a: Fisher information of the Rakhmanov density associated to the orthonormal Gegenbauer

polynomials C(λ)n (x); dependence with n and λ. b: Distributions for n = 2 and λ = 0.1,1.1, where the Fisher

information has the largest value. One observes that the two distributions are qualitatively different. c:

Conversely, the Fisher information has a very low value for n = 0 and λ = 9.5,10.5, and the distributions canbe seen to be, in fact, qualitatively similar.


by a plateau of the Fisher information). Again, an evaluation of the Kullback-Leibler divergencecan be done, obtaining D [ρ2(x; 0.1)∥ρ2(x; 1.1)] = 0.85 and D [ρ0(x; 9.5)∥ρ0(x; 10.5)] = 0.0025, forthe distributions depicted in Figs. 3.2b and c, respectively.

This work, together with Ref. [31], opens the way for the developement of the Fisher estimationtheory of the Rakhmanov density for continuous and discrete orthogonal polynomials in and beyondthe Askey scheme. This fundamental task in approximation theory includes the determination of thespreading of the orthogonal polynomials throughout its orthogonality domain by means of the Fisherinformation with a locality property. All these mathematical questions have a straightforwardapplication to quantum systems because their wavefunctions are often controlled by orthogonalpolynomials, so that the probability densities which describe the quantum-mechanical states of thesephysical systems are just the Rakhmanov densities of the corresponding orthogonal polynomials. Inparticular, they correspond to the ground and excited states of the physical systems with an exactlysolvable spherically symmetric potential [64], including the most common prototypes (harmonicoscillator and hydrogen atom), in both position and momentum spaces.

Most of the results presented in this chapter can be found published in Ref. [72].

Part II.

Collective Rydberg excitations of an

atomic gas confined in a ring lattice

4. Introduction

Nowadays, the laser and evaporative cooling of atoms to temperatures below one micro-Kelvinare well-established experimental techniques, that have opened a doorway to a wide range of newexperiments with ultracold atoms. In particular, the achievement in 1995 of the Bose-Einsteincondensation in atomic gases of rubidium, sodium and lithium [73, 74, 75], that had been theoret-ically predicted more than 80 years ago, can be set as the starting point of this new development.Fermionic gases were also brought to degeneracy only a few years later [76].

Ultracold atoms provide a unique toolbox to study many-particle physics under very clean andwell-defined conditions. The reasons for this are rooted in: (i) the precise control that can beachieved over the interactions in cold gases using Feshbach resonances, and (ii) the possibility ofgenerating strong external trapping potentials for cold atoms through optical, magnetic or electricfields. These features allow to study the dynamics of phase transitions as well as the preparationof strongly correlated quantum states (see [77] for a complete overview in this topic).

While so far the majority of experiments in ultracold gases is carried out with ground stateatoms, during the recent years an increasing amount of experimental and theoretical efforts havebeen dedicated to the study of atoms excited to Rydberg states [78]. This is due to the uniqueproperties of these highly excited states. In particular, Rydberg atoms can interact via dipole-dipoleor van-der-Waals forces with interaction strengths that can be of the order of several tens of MHz ata distance of several micrometers. As a consequence, the corresponding quantum dynamics takesplace on a microsecond timescale, which is orders of magnitude faster than the atoms’ externaldynamics. Such scenario is usually referred to as ’frozen gas’ [79, 80] and the evolution of Rydbergexcitations is usually described by a spin model [81, 82], where the spin up/down state representsa Rydberg/ground state atom. Unlike in a typical solid state system, there are no significantdissipative processes which make the system assume its ground state over the typical experimentaltimescale. Therefore, in this kind of systems one can regard the dynamics as fully coherent [83, 84]and the time evolution of quantities like the mean number of Rydberg excitations is expected todepend crucially on the initial state.

The most intriguing manifestation of the strong interaction among Rydberg states is the block-ade mechanism [85, 86], which prevents the excitation of a Rydberg atom in the vicinity of analready excited one. This effect has been thoroughly studied in the context of quantum infor-mation processing since it is a natural implementation of a state dependent interaction which isessential to devise two-qubit gates. In the context of gases, a first experimental indication of thestrong Rydberg-Rydberg interaction was the non-linear behavior of the number of excited atomsas a function of increasing laser power and atomic density [87, 88]. Later, it was shown that, inthe case of a dense gas, the Rydberg blockade gives rise to the formation of coherent collectiveexcitations - so-called ’superatoms’ [89, 90]. Very recently, the power of Rydberg states to establisha controlled interaction of single atoms confined in distant traps has been demonstrated in a seriesof impressive experiments [91, 92, 93, 94].

Strongly supported by these results, nowadays highly excited atoms are believed to have a man-ifold of applications ranging far beyond traditional atomic physics. On the theory side, it hasbeen shown that the long-ranged character of the interaction can be employed to manipulate wholeatomic ensembles by just a single control atom [95]. Moreover, exploiting the properties of atoms inRydberg states permits the study of spin systems at criticality [82], the quantum simulation of com-plex spin models [96], the investigation of the thermalization of strongly interacting many-particle

36 Introduction

systems [97] and also the implementation of quantum information protocols [98].In the present work, we study the excitation properties of a Rydberg gas in a particularly struc-

tured and symmetric scenario (see Chapter 6). In our setup, ground state atoms are homogeneouslydistributed over a ring lattice and at most a single Rydberg atom per site can be excited via alaser. In Chapter 5, some preliminary concepts needed for the understanding of this part of thethesis are introduced, such as the atom-light interaction or the Rydberg atoms. That chapter can,hence, be skipped by the reader who is familiar with quantum optics and Rydberg physics.

In the framework of the perfect blockade regime, in Chapter 7 we study the temporal evolutionof the Rydberg excitation number, the formation of correlations in the Rydberg density and theentanglement properties in lattices with up to 25 sites. We demonstrate that the dynamics of thissystem is divided into short and long time domains and that in the latter the system acquires asteady state. We study the origin of this steady state which occurs as result of a purely coherentdynamics of a closed system in Chapter 8. This subject is closely related to the discussion abouthow and why a closed system which is prepared in a pure state actually thermalizes, i.e., assumesa state in which the mean values of macroscopic observables are stationary and can be calculatedfrom the microcanonical ensemble. These questions are of fundamental interest [99, 100] and havebeen investigated in a number of systems [101, 102, 103, 104, 105].

We also show (Chapter 9) that the setup mentioned above allows to create and explore collectivemany-particle states that are entangled and extend over the entire lattice, e.g., spin waves, ona microsecond timescale. Finding simple ways for creating entangled many-particle states is ofimportance, since such states have a number of applications, e.g., they serve as resource for thecreation of single-photon light sources [106], for improving precision quantum measurements [107]and for measurement-based quantum information processing. We finally show that the systemoffers the possibility to study fermions in the presence of a disorder potential although no externalatomic motion takes place.

Most of the results presented in this part can be found in the following publications: B. Olmos, R. Gonzalez-Ferez and I. Lesanovsky, ”Collective Rydberg excitations of an atomicgas confined in a ring lattice”, Phys. Rev. A, vol. 79, p. 043419, 2009. B. Olmos, R. Gonzalez-Ferez and I. Lesanovsky, ”Fermionic collective excitations in a latticegas of Rydberg atoms”, Phys. Rev. Lett., vol. 103, p. 185302, 2009. B. Olmos, M. Muller and I. Lesanovsky, ”Thermalization of a strongly interacting 1D Rydberglattice gas”, New J. Phys., vol. 12, p. 013024, 2010. B. Olmos, R. Gonzalez-Ferez and I. Lesanovsky, ”Creating collective many-body states withhighly excited atoms”, Phys. Rev. A, Accepted, arXiv: 0812.4894, 2010.

5. Preliminary concepts

This chapter has an introductory character, since in it some preliminary concepts are presented.Throughout all this work we consider trapped atoms whose internal states (ground and excited) arecoupled by laser light. Hence, the form of the interaction between a two-level atom and a laser fieldis derived. Then, we use the second quantization picture to obtain a Hamiltonian that describesthe dynamics of this coupling in terms of creation and annihilation operators of ground and excitedatoms. The excited level of the atoms that we consider here is a so-called Rydberg state. Hence,the properties of these Rydberg states are also reviewed in this chapter, paying special attentionto the strong interaction between them and its consequences.

5.1. Interaction of a two-level atom and a monochromatic light field

We consider a single atom in a trap, represented by the external potential U(R) with R being thecenter of mass coordinate. The Hamiltonian that drives the external dynamics of this system isgiven by the sum of the kinetic and potential energies, i.e.,

Hext = P2

2M+U(R),

where M is the mass of the atom and P represents its momentum. The eigenfunctions and eigen-values of this Hamiltonian are Hextφλ(R) = ǫλφλ(R), with λ denoting the mode of the trappingpotential. We consider that the atom has only two internal levels ∣g⟩ and ∣r⟩. The energy of thestate ∣g⟩ is put to zero, so the energy of ∣r⟩ corresponds to the gap between the two levels ωa (inatomic units). The internal dynamics is then described by the Hamiltonian

H0 = ωa ∣r⟩ ⟨r∣ .We introduce now a laser field as a plane wave with momentum k and frequency ωl as

E(r, t) = E0 cos(ωlt − k ⋅ r)ε,where E0 is the amplitude of the light field and ε is the unit polarization vector. The laser isdetuned ∆ = ωa − ωl with respect to the transition ∣g⟩ → ∣r⟩ (see Fig. 5.1). We work here withinthe dipole approximation, i.e., we consider that the field does not vary over the extension of theatom. In that case, one can describe the interaction of this laser field with the atom by means ofthe Hamiltonian

Hlaser = −r ⋅E(R, t),r

g

wa wl

D

Figure 5.1.: Atom modeled as a two-level system. These states are coupled by means of a laser field withfrequency ωl in general not resonant with the energy gap ωa, i.e., with a detuning ∆ = ωa − ωl.

38 Preliminary concepts

where, as pointed out before, R is the external position of the center of mass of the atom andr ≡ (x, y, z) accounts for the relative coordinate of the electron. If we consider the laser to belinearly polarized in the direction ε = −z, then the laser Hamiltonian becomes

Hlaser = E0z cos(ωlt − k ⋅R).The complete Hamiltonian that drives the dynamics of the two-level atom coupled to a monochro-

matic light field is thus given by the sum of the three terms,

H =Hext +H0 +Hlaser.

We can rewrite this Hamiltonian as a matrix representing it in the basis composed of the two

possible internal states of the atom ∣g⟩ ≡ ( 01) and ∣r⟩ ≡ ( 1

0) as

H = ( Hext + ωa +E0 ⟨r∣ z ∣r⟩ cos(ωlt − k ⋅R) E0 ⟨r∣ z ∣g⟩ cos(ωlt − k ⋅R)E0 ⟨g∣ z ∣r⟩ cos(ωlt − k ⋅R) Hext +E0 ⟨g∣ z ∣g⟩ cos(ωlt − k ⋅R) ) ,

where we have assumed Hext to be independent of the internal state of the atom. The matrixelements ⟨r∣ z ∣r⟩ and ⟨g∣ z ∣g⟩ vanish since ∣g⟩ and ∣r⟩ are parity eigenstates and the cross terms⟨r∣ z ∣g⟩ = ⟨g∣ z ∣r⟩ are absorbed in the so-called Rabi frequency

Ω0 ≡ E0

2⟨r∣ z ∣g⟩ , (5.1)

so that

H = ( Hext + ωa 2Ω0 cos(ωlt − k ⋅R)2Ω0 cos(ωlt − k ⋅R) Hext

) .To eliminate the time-dependence of the Hamiltonian, we go now into a rotating frame by means

of an unitary transformation

U = ( e−iωlt 00 1

) .This operation transforms the time-dependent Schrodinger equation such that

i∂

∂tΨ =HΨ → i

∂

∂tΨ = HΨ,

with

Ψ = U Ψ H = U HU − iU ∂∂tU,

so the Hamiltonian is transformed such that

H ≈Hext +∆ ∣r⟩ ⟨r∣ + (Ω0reik⋅R ∣r⟩ ⟨g∣ +Ω0re

−ik⋅R ∣g⟩ ⟨r∣) (5.2)

where we have made the Rotating Wave Approximation (RWA) neglecting the terms oscillatingwith a frequency 2ωl.

5.2 Second quantization 39

5.2. Second quantization

We are now interested in deriving the Hamiltonian that drives the dynamics of the atom-lightinteraction in terms of the creation and annihilation operators of the ground and excited-stateatoms. To do so, we go to the second quantization picture, and introduce the field operators thatannihilate a particle in a given internal state at R in the λ-th mode of the trapping potential as

ψ(R) = ( ∑λ aλ(1)φλ(R)∑λ′ aλ′(2)φλ′(R) ) ≡ ( ψ1(R)

ψ2(R) ) = ψ1(R) ∣r⟩ + ψ2(R) ∣g⟩ .The components of the field operators obey the commutation relation

[ψk(R), ψk′(R′)] = δkk′δ(R −R′),

given that the creation and annihilation operators satisfy the usual bosonic commutation relations

[aλ(k), aλ′(k′)] = δλ,λ′δkk′, [aλ(k), aλ′(k′)] = [aλ(k), aλ′(k′)] = 0,and that the functions φλ form an orthogonal set. Here, we denote by aλ(1) ≡ rλ and aλ(2) ≡ bλthe annihilation operators of an atom in the excited and ground state, respectively, and φλ(R) arethe eigenfunctions of Hext. The Hamiltonian (5.2) can be rewritten in terms of these field operatorsas follows

H = ∫ dRψ(R)Hψ(R) =∑λ

ǫλ (bλbλ + rλrλ) +∆∑λ

rλrλ +Ω0∑

λ

(bλrλe

ik⋅R + rλbλe−ik⋅R) ,

where we have assumed that the external potential is independent of the internal state of the atom.Taking into account that we consider the atoms lying in the ground state of the trapping potential(λ = 0), and setting this ground state energy to zero (ǫ0 = 0), we obtain

H =∆rr +Ω0 (breik⋅R + rbe−ik⋅R) .One can always find another unitary transformation U2 to remove the dependence on the positionof the atom such that

U2rU2 = e−ik⋅Rr, U

2r

U2 = eik⋅Rr,and, thus, the Hamiltonian yields

H =∆rr +Ω0 (br + rb) . (5.3)

Note that the wavefunction acquires then a phase that one has to take into account in the case ofmore than one atom, e.g., when measuring correlations between several atoms in different positions.This Hamiltonian can be easily extended to its many-particle version, as will be done in Section6.1.

5.3. Rydberg atom

A Rydberg atom is an atom in a state with very high principal quantum number n ≫ 1 that, asa consequence, possesses exaggerated properties [78]. Historically, the first time that this conceptappeared was in 1885, with Balmer’s formula for the wavelengths of the visible lines of the hydrogen

λ = bn2(n2 − 4)2 .


−9

−8.8

−8.6

−8.4

−8.2

−8x 10−3

Ene

rgy

(eV

)

39 39

40 40

41 41

41d

42p

42d

43s

43p

44s

Hydrogen Rubidium

Figure 5.2.: Scheme of the spectra of the Rydberg states of a hydrogen and a rubidium atom. Due to thequantum defect the spectrum decays into degenerate n-manifolds (hydrogen-like) except for the low orbitalquantum number states, that are energetically isolated from the rest.

We now know that this expression, with b = 3645.6 A, gives the wavelengths of the transitions fromthe state of hydrogen with n = 2 to higher lying levels, as can be seen rewriting it as

ν = 1

4b(14− 1

n2) .

With the theory of Bohr of the atomic structure of the hydrogen atom in 1913, the physicalsignificance of the states with high principal quantum number n was understood for the first time.In his picture, the hydrogen atom was constituted by a proton and an electron describing orbitsaround it. The orbits allowed for the electron (in which it would not emit radiation) were given bymultiples of the orbital momentum. With this simple model, in particular, an expression for theorbital radius of a hydrogen atom in a given state was obtained to be proportional to n2,

r = a0n2,with a0 = 0.529 A being the so-called Bohr radius. Also, it was found that the binding energy ofthe electron decreases as n−2

W = −Ryn2,

where Ry = −13.6 eV is the universal so-called Rydberg constant. The value of this constant hadalready been measured in the year 1890 by J.R. Rydberg when he began to classify the spectra ofalkali atoms into sharp, principal and diffuse series of lines (s, p and d states, respectively, notationthat survived until nowadays). As a consequence, from Bohr’s theory one could physically under-stand a Rydberg atom as one whose valence electron is in a very large and loose orbit characterizedby the quantum number n. We know now that this description is correct from the intuitive point ofview but not complete. To actually obtain a description of the Rydberg atoms, quantum mechanicsis necessary. The wavefunction ψ, solution of the Schrodinger equation

(−∇2

2− 1

r)ψ =Wψ, (5.4)

will provide us the different properties of the corresponding system. The solution is of the form

ψnlm(r) = CRnl(r)Ylm(θ,φ), (5.5)

5.4 Interacting Rydberg atoms 41

with C being the normalization constant, and where Ylm(θ,φ) stands for the so-called sphericalharmonics. Note that the dependence of the wavefunction with the principal quantum number isencoded exclusively in the radial part Rnl(r).

The description of the alkali Rydberg atoms is parallel to the hydrogen case, and the generalform of the wavefunctions is very similar. The difference between the two cases is that in the alkaliatom the positive core, although having a net charge of +1, is formed by a nucleus with charge+Z and Z − 1 electrons, i.e., has structure and cannot be considered ’point-like’ as the proton. Inhigh angular momentum states l ≥ 4, the basic properties of the two systems are the same, sincethe valence electron in this kind of orbit does not ’see’ the inner structure of the core. On theother hand, when l takes low values, as in the case of the s, p or d states, the valence electronpenetrates the inner core and, as a consequence, the corresponding wavefunctions of the alkaliatoms differ from their hydrogen counterparts. Hence, the energy levels of an alkali Rydberg atomare accurately described by

Enl = − Ry(n − δl)2 = −Ry

n∗2,

where, δl stands for the so-called quantum defect, a function of the orbital quantum number andn∗ = n − δl is the effective principal quantum number. In the following, when we mention n wewill be actually referring to n∗. The quantum defect is negligible except in the case of low l,particularly high for s-states, so that the spectrum decomposes into degenerate manifolds with thesame principal quantum number n except for these low-l states (see Fig. 5.2 to see the case of therubidium atom). In this work, when we use Rydberg states, we shall focus on ns-states that areenergetically well isolated from the rest.

Property n dependence

Orbital radius n2

Binding energy n−2

Radiative lifetime n3

Energy level spacing n−3

Dipole matrix elements n2

Polarizability n7

Table 5.1.: Dependence of several features of Rydberg atoms with the (effective) principal quantum numbern.

Other properties of these Rydberg states can be inferred from the wavefunction solution of thecorresponding Schrodinger equation. In particular, their scaling with the principal quantum numbern are given in Table 5.1, extracted from Ref. [78]. Let us highlight that the lifetime of these highlyexcited states is rapidly increasing with n, τ ∝ nα. The exponent α changes with the orbitalquantum number l, being α = 3 for s-states and α = 5 for circular states, i.e., l = n − 1. To have anidea of the timescale we are referring to, the lifetime for Rb in the 43s state is approximately 90µs [78]. Another interesting feature of these highly excited states is their large polarizability. It isobtained by the sum of the squared matrix elements between the state and the neighboring levels⟨nl∣ r ∣n′l′⟩, that scales as n2, divided by the energy difference between the levels, given roughly byn−3. As a consequence the corresponding polarizability scales as n7, which gives very large valuesthat yield to very strong interactions between the Rydberg atoms.


R12

r2

r1

--

+ +

Figure 5.3.: Scheme of two Rydberg atoms separated by a large distance R12.

p1

p´1Ep´

Ep

0 s1

p2

p´2Ep´

Ep

0 s2

Figure 5.4.: Scheme of the spectrum of two Rydberg atoms in the s-state, whose energy is put to zero.

5.4. Interacting Rydberg atoms

The interaction energy between two Rydberg atoms separated by a very large internuclear distanceR12 = R12z in comparison with the extension of the atoms (see Fig. 5.3) can be written as

V12(R12) = 1

R312

[r1 ⋅ r2 − 3 (r1 ⋅ z) (r2 ⋅ z)] .In order to analyze the interaction properties of the Rydberg atoms and obtain, in particular, thedependence of its strength with the principal quantum n, we make use of a simplified model. Formore elaborate calculations, see Refs. [108, 109].

5.4.1. Simple model

Let us consider the situation where two atoms are in a certain ns state, as it is outlined in Fig.5.4, i.e., the initial situation is ∣S⟩ ≡ ∣s1⟩⊗ ∣s2⟩. Given the form of the wavefunctions (5.5) and theproperties of the spherical harmonics, it is easy to see that this state is only coupled by means ofthe interaction to the levels where both of the atoms are in a p-state (note that we put here theenergy of the s-state to zero, so that Ep > 0 and Ep′ < 0). Thus, those most strongly coupled to theinitial one, ∣S⟩, are

∣P ⟩ ≡ ∣p1⟩⊗ ∣p2⟩ , ∣P ′⟩ ≡ ∣p′1⟩⊗ ∣p′2⟩ , ∣PP ′⟩ ≡ 1√2[∣p1⟩⊗ ∣p′2⟩ + ∣p′1⟩⊗ ∣p2⟩] .

We consider a regime in which Ep + Ep′ ≪ 2Ep,Ep′ , i.e., the first two states ∣P ⟩ and ∣P ′⟩ are fardetuned and thus we take the corresponding probabilities of transition to those to be negligible.Within this approximation, the Hamiltonian H =H1 +H2 + V12 can be written in terms of the twomost strongly coupled states ∣S⟩ and ∣PP ′⟩

H ≈ ⎛⎝0 α

R312

αR3

12

Ep +Ep′

⎞⎠ ≡ ( 0 V

V δ) ,

5.4 Interacting Rydberg atoms 43

gr

W0 r r

wgr

wgr

gg

rg

R12

E

rB

C /R6 12

6

Figure 5.5.: Scheme of the Rydberg blockade mechanism. The ratio between the Rabi frequency Ω0 and thevan-der-Waals coefficient C6 determines the blockade radius rB from which no double excitation is accessibleby the laser field.

where ⟨S∣r1 ⋅ r2 − 3 (r1 ⋅ z) (r2 ⋅ z) ∣PP ′⟩ ≡ α. It can be easily diagonalized obtaining as a result theenergies

E± = δ2

⎡⎢⎢⎢⎢⎣1 ±√

1 + (2Vδ)2⎤⎥⎥⎥⎥⎦ ≈

δ

2± δ2± V

2

δ,

where we have considered that δ ≫ V . Hence, the two ns Rydberg atoms interact via the quicklydecaying van-der-Waals potential

VvdW(R12) = −V 2

δ≡ − C6

R612

,

with the van-der-Waals coefficient being C6 = − α2

Ep+Ep′∝ n11. The dependence of C6 with n can

be obtained considering that in this simple model we do not take into account the specific angularconfiguration of the atoms, and thus α ∼ ⟨S∣ r1 ⋅ r2 ∣PP ′⟩ ∝ n4, and that the energy separation isproportional to n−3, as given in Table 5.1.

5.4.2. Rydberg blockade

Given the dependence of C6 ∝ n11, one can see that the interaction between Rydberg states isvery strong. For example, the van-der-Waals coefficient for the 43s state of rubidium is given byC6 = −2.4510−27 MHz ⋅m6 [108]. Hence, two of those Rydberg atoms placed 3µm apart interactwith a strength of 3.35MHz.

As a consequence of this strong interaction, the so-called blockade effect arises [85, 86]. Let usconsider two atoms at a distance R12 from each other that are resonantly driven from the groundstate ∣g⟩ to a Rydberg state ∣r⟩ via a laser field. The possible states of this two-body system are thus∣gg⟩ , ∣gr⟩ , ∣rg⟩ and ∣rr⟩, as depicted in Fig. 5.5. Note that the energy needed to excite both states∣gr⟩ and ∣rg⟩ is the same, ωgr. When the interparticle distance is very large, as a consequence of thequickly decaying character of the van-der-Waals interaction, the atoms in the Rydberg state ∣rr⟩do not interact, and thus its energy is given by two times the single-excitation one, 2ωgr. Hence,in this regime the laser can excite both atoms. As the interparticle distance becomes smaller theinteraction becomes stronger and thus the energy of the doubly excited state is shifted, until thelaser field is no longer capable of exciting it. In the simplest model, the so-called blockade radius,i.e., the distance at which we consider the laser out of resonance, can be estimated by equaling


the van-der-Waals interaction with the power-broadened linewidth of the laser given by its Rabifrequency (proportional to its intensity) [89], i.e., r6B ∝ C6/Ω0.

In summary, the effect of the interaction between the Rydberg states is a suppression of theexcitation of more than one atom in a neighborhood defined by the ratio between the Rabi frequencyof the laser and the interaction strength. As a consequence, within this blockade radius there canonly be one excitation, which in the case of two atoms corresponds to a symmetric superposition ofthe two possible excited states, i.e., [∣gr⟩ + ∣rg⟩] /√2. Extending this to a gas of atoms yields to theconcept of ’superatom’ (see Refs. [89, 90]), that is nothing but a collective delocalized excitationin an area (for a 2D gas) or volume (3D) defined by the blockade radius rB . This concept will beexplained in more detail in the next section and used extensively throughout all this work.

6. Laser-driven atomic gas confined to a ring

lattice

In this chapter we use the concepts introduced in the previous one to present the system that isthe focus of our study, i.e., an atomic gas of atoms confined to a lattice whose internal states arecoupled by means of a laser field. We consider also the interaction between the atoms and derivethe Hamiltonian that drives the dynamics of this system. Finally, we explain how the symmetriesof the setup can be exploited in order to simplify the problem.

6.1. The system and its Hamiltonian

We study a gas of bosonic ground-state atoms confined to a deep large spacing ring lattice withperiodicity a ≈ µm (see Fig. 6.1). These ring lattices can be created approximatively starting from alarge spacing standard rectangular lattice [110], removing then the atoms from the unwanted sites,thereby ’cutting out’ a ring. It has been experimentally demonstrated that this is possible startingfrom magnetic arrays of microtraps on a chip and using a focused laser to address and rapidly emptyindividual traps [111]. Optical 2D lattices have been also shown to be a good candidate to create aring lattice, where the individual sites are here addressed and emptied by the use of electron beams[112]. Near the lattice minima, the atomic motion approaches to that of the harmonic oscillator.For sufficiently large trap depths, the atoms are localized near the potential minima. Each site hasthus an harmonic oscillator wavefunction associated with it, and for sufficiently strong potentials,there is very little overlap between the wavefunctions φk(x) in different wells. In other words, weconsider the functions φk(x) of the atoms lying on the k-th site to be very narrow, i.e., localizedwith a width σ ≪ a.

The internal (electronic) dynamics - in which we are interested here - takes place on a muchshorter timescale than the external dynamics (hundreds of nanoseconds and milliseconds, respec-tively). Hence, we can assume the external dynamics of the atoms to be frozen, i.e., no hoppingand thus no particle exchange between the lattice sites is present.

As in Sections 5.1 and 5.2, we consider only two internal electronic levels in each atom, which aredenoted by ∣g⟩ and ∣r⟩. Here, ∣g⟩ stands for the ground state and ∣r⟩ for a highly excited (Rydberg)ns-state which - as we explained in Sec. 5.3 - is well isolated from any other electronic level.We assume that the atoms experience the same trapping potential independently of their internalstate. This requirement is, however, not crucial as the typical timescale of the electronic excitationdynamics, which we are going to study, is much smaller than the dephasing time due to differenttrapping potentials experienced by ∣g⟩ and ∣r⟩. Experimentally these two levels are usually coupledby a two-photon transition (see Appendix A). Here, we assume, without any loss of generality, thatthey are coupled resonantly by a laser of Rabi frequency Ω0 and detuning ∆.

The system can be considered, then, as an ensemble of two-level atoms lying in L independentharmonic traps. Hence, the coupling of these atoms with the laser field can be described (withinthe RWA) as an extension of the Hamiltonian derived in Sec. 5.2, i.e.,

HL = Ω0

L

∑k=1

(bkrk + rkbk) +∆ L

∑k=1

nk, (6.1)

46 Laser-driven atomic gas confined to a ring lattice

D

|Pñk

W

|Rñk

a

s k+1

k-1

k

Figure 6.1.: Ring lattice with spacing a being much larger than the extension σ of the external wavefunctions(deep lattice). The internal atomic degrees of freedom at each site are described by the (collective) states∣P ⟩k and ∣R⟩k, coupled by Ω.

where bkand r

k(bk and rk) represent the creation (annihilation) of a ground and a Rydberg state,

respectively, and nk = rkrk stands for the number of atoms in state ∣r⟩ at the k-th site. For thesake of simplicity, we will consider in the following the case where each lattice site is occupied bythe same number of atoms, N0. This is achieved, for example, if the system is initialized in aMott-insulator state.

The interaction between the Rydberg atoms is given by the van-der-Waals potential V (x) =−C6/x6, that is quickly decaying with the distance x between excited atoms. Nevertheless, as C6

scales with the eleventh power of the principal quantum number n, the interaction can stronglyaffect the excitation dynamics of atoms that are separated by several micrometers. This stronginteraction gives rise to the so-called blockade effect [85, 86] (see Sections 5.3 and 5.4). We considerthat the simultaneous excitation of two or more atoms to the Rydberg state on a single lattice siteis blockaded. This is justified since one can always find a scenario (large enough n and/or deepenough lattice) in which the corresponding blockade radius is larger than the extension σ associatedto the atoms in a site. Thus, on each lattice site k, only the two states

∣P ⟩k = [∣g⟩k]1 ⊗ . . . [∣g⟩k]N0

∣R⟩k = 1√N0

S [∣r⟩k]1 ⊗ [∣g⟩k]2 ⊗ . . . [∣g⟩k]N0 ,

are accessible, where S represents the symmetric superposition operator. ∣P ⟩k is the product statein which all the atoms on site k are on the ground state. On the other hand, ∣R⟩k is an extension ofthe two-atom case we considered in Section 5.4, a symmetric superposition of the N0 possible statesin which one atom is excited. Thus, it describes a delocalized collective excitation all over the site,known in the literature as ’superatom’ [89, 90]. The effective Rabi frequency for the laser couplingbetween these two collective states (see Fig. 6.1) is given by k ⟨R∣HL ∣P ⟩k = k ⟨P ∣HL ∣R⟩k = Ω0

√N0 ≡

Ω. Taking all this into account, in Eq. (6.1) we can replace Ω0 rkbk → Ωσ

(k)+ = (Ω/2)[σ(k)x + iσ(k)y ],

where σ(k)i are the Pauli spin matrices:

σx = ( 0 11 0

) σy = ( 0 −ii 0

) σz = ( 1 00 −1 ) ,

so the laser Hamiltonian can be rewritten as

HL = Ω L

∑k=1

σ(k)x +∆L

∑k=1

nk. (6.2)

Since we consider the extension of the external wavefunctions to be much smaller than the latticespacing, we can assume φk(x) ≈ δ(x−Pk) (with Pk representing the coordinates of the center of the

6.2 Analysis of the symmetries 47

site k) and rewrite the van-der-Waals potential between two (super)atoms in the state ∣R⟩ locatedat the sites i and j, as

V∣i−j∣ = ∫ dx1dx2 ∣φi(x1)∣2 ∣φj(x2)∣2 V (∣x1 − x2∣) ≈ V (∣Pi −Pj ∣).Hence, the interaction yields Vd = −C6/x6d, where xd is the separation between the (super)atomslocated d sites apart, and the interaction Hamiltonian reads

Hint =

L

∑k=1

d

∑l=1

Vlnknk+l,

with the Rydberg number operator nk = [1+σ(k)z ]/2 and the boundary condition σ(1)j = σ

(L+1)j (ring

lattice). When the range of the interactions is much smaller than the ring size, one can approximatethe van-der-Waals potential as Vd = −C6/(ad)6. Thus, the interaction Hamiltonian can be rewrittenin terms of the nearest neighbor interaction β ≡ V1 = −C6/a6 as

Hint = βL

∑k=1

d

∑l=1

nknk+l

l6.

Summarizing, the complete Hamiltonian that drives the dynamics of a frozen laser-driven Ryd-berg gas confined to a monodimensional ring lattice can be written as

H =L

∑k=1

[Ωσ(k)x +∆nk + βd

∑l=1

nknk+l

l6] . (6.3)

Hence, the relevant parameters in our system will be: (i) the ones related to the laser, i.e., the single-atom Rabi frequency Ω0 and detuning ∆, which can be time-dependent and (ii) the interactionstrength between Rydberg atoms represented by β and its maximal range, da. Throughout thiswork, we consider the regime where the detuning is much smaller than both the collective Rabifrequency and the interaction strength, i.e., ∣∆∣ ≪ Ω, β. As a consequence, the behavior of thesystem will be determined by the strength ratio of the laser and the interaction. In particular,we study here the two limiting cases. In Chapters 7 and 8, we consider that the interaction ismuch stronger than the laser field, so that the double excitation is forbidden in neighboring sites(perfect blockade). In Chapter 9, the regime where the laser field is much stronger than the nearestneighbor interaction, Ω≫ β, is studied.

6.2. Analysis of the symmetries

Our goal is to study the dynamics of the system under the action of Hamiltonian (6.3) from agiven initial state. To this end, first we take into account the symmetries of the system and itsHamiltonian, since they lead to a significant simplification of the problem when solving the time-dependent Schrodinger equation.

There are two basic symmetries on a ring lattice which are of interest in our system: cyclic shiftsby l sites and the reversal of the order of the lattice sites. The former is represented by the unitaryoperators Xl with l = 1,2, . . . ,L where Xl = X

l1, while the latter operation is the parity and is

denoted by R. The action of these operations on the spin ladder operators σ(k)± is defined through

Rσ(k)+ R = σ(L−k+1)+ R

σ(k)− R = σ(L−k+1)−

Xlσ(k)+ Xl = σ

(k+l)+ X

lσ(k)− Xl = σ

(k+l)− ,

48 Laser-driven atomic gas confined to a ring lattice

from where follows that the Hamiltonian (6.3) is invariant under these operations ([H,Xl] =[H,R] = 0). In other words, both shift and reversal operations correspond to conserved quan-tities. Our complete Hilbert space is in principle spanned by all the possible configurations in thering, taking into account that each site can be only in two states, ∣P ⟩ or ∣R⟩. Thus, for an increasingsite number L, the exact solution of this problem becomes quickly intractable as the dimension ofthe total Hilbert grows as 2L. However, if the system is initialized in an eigenstate with respectto Xl and R, the time evolution will not take place in the entire Hilbert space, but merely in thesubspace spanned by the states with the same quantum number with respect to Xl and R.

In practise, the natural initial situation will be that in which all atoms are in the ground state,i.e., all sites in the product state ∣P ⟩,

∣0⟩ = L

∏k=1

∣P ⟩k .Studying the symmetry properties of this initial state where no Rydberg atom is excited tells usthat it is an eigenstate of all cyclic shifts and the reversal operator with eigenvalue 1:

Xl ∣0⟩ = ∣0⟩ R ∣0⟩ = ∣0⟩ .We will refer to such a state that has eigenvalue 1 with respect to Xl andR as being fully-symmetric.Only a small subset of the 2L states spanning the whole Hilbert space actually has these properties.In particular, each of these fully-symmetric states can be understood as a superposition of all statesthat are equivalent under rotation and reversal of the sites. For example, when one only (super)atomis excited on the ring, the corresponding fully-symmetric state (denoted by the subscript S) reads

∣1⟩S = 1√LS ∣R⟩1 ⊗ ∣P ⟩2 ⊗ . . . ∣P ⟩L .

In summary, only the states from this fully-symmetric subspace can be actually accessed in thecourse of the system’s time evolution under Hamiltonian (6.3).

The dimension of the problem is like this dramatically reduced. To pick an example, for L = 10 thedimension of the Hilbert space decreases from 210 = 1024 down to 78. For our computations we needan algorithm to quickly generate the maximally symmetric states among which the evolution takesplace. Such an algorithm is discussed in Appendix D.1. There, these states are called bracelets,and are recursively generated in an optimal way. The amount of CPU time grows only proportionalto the number of bracelets produced [113].

7. Strong interaction: Perfect blockade

A great simplification of the Hamiltonian (6.3) is achieved in the so-called perfect blockade regimewhich we are going to introduce and analyze here. Within this regime, we study numericallythe time evolution of the system through different local properties as the Rydberg density, thecorrelation functions and the entanglement. In the last section of the chapter we relax the perfectblockade condition and test like this the validity of our approximations.

7.1. Perfect blockade regime

As already pointed out, the interaction between Rydberg atoms is quickly decaying with the dis-tance. In particular, the next-nearest neighbor interaction is a factor of 64 cos6 (π/L) smaller thanthe nearest neighbor one (V2 ≪ β). At the same time, here we are going work in a regime wherethe laser field strength given by the collective Rabi frequency is much stronger than the next near-est neighbor interaction, i.e., Ω ≫ V2. As a consequence, we will thus consider that only nearestneighbor interaction takes place. The Hamiltonian for the entire atomic ensemble then reads

H =L

∑k=1

[Ωσ(k)x +∆nk + βnknk+1] . (7.1)

Thus, the system can be described as a periodic arrangement of spin-1/2 particles, where the twospin states, corresponding to the two internal states of the (super)atoms, ∣P ⟩k and ∣R⟩k, interactvia an Ising-type potential. In this picture, the Rabi frequency Ω and the combination of ∆ + βcan be effectively interpreted as perpendicular magnetic fields.

Initially, the laser is turned off Ω(0) = 0. Thus, the spectrum of the Hamiltonian H is that of theinteraction Hamiltonian Hint = β∑L

k=1 nknk+1. It is composed by L degenerate subspaces spannedby states that have the same number of pairs of neighboring excitations ν = 0,1, ...,L − 2,L, withcorresponding energy Eν = ν β, (note that the subspace ν = L − 1 is not included because of theimpossibility of having L − 1 pairs of neighboring excitations in a ring).

HL HL

Dn=0 |Dn|=1 |Dn|=2

a

n

n+1

n+2 HL

b

b

HLHL

HL

HL

HL

b

Figure 7.1.: a: The action of the laser creates or annihilates an excitation ∣R⟩ (red circles), that gives riseto a difference of pair of excitations ∆ν = 0,±1,±2 depending on the neighboring sites. b: Energy levelstructure of the Hamiltonian (6.3) with β ≫ Ω. The spectrum consists of highly degenerate subspaces whichare labeled by ν energetically separated by β, except the levels with ν = L−2 and ν = L, which are separatedby 2β (see text for explanation). The laser (HL) causes an energy splitting of the degenerate levels. Inaddition, it couples states belonging to a given ν-subspace and connects subspaces with ∣ν − ν′∣ = 1,2.

50 Strong interaction: Perfect blockade

Let us discuss the action of the laser Hamiltonian HL, that is switched on instantaneously at agiven time t0 = 0. Its operation creates or destroys a (super)atom state ∣R⟩k on the k-th site ofthe ring. This leads to three different situations depending on the state of the neighboring sites: i)both in the state ∣P ⟩, ii) one in ∣P ⟩ and one in ∣R⟩, or iii) both in state ∣R⟩ (see Fig. 7.1a). Thedifference of number or pairs of neighboring excitations is ∆ν = 0,±1,±2, respectively. Thus, HL

drives the dynamics within a given ν-subspace and couples subspaces with ∣ν − ν′∣ = 1,2.We consider now that β ≫ Ω (see Fig. 7.1b), i.e., the interaction energy of two neighboring

Rydberg atoms is larger than the collective Rabi frequency. Let us point out that in this limit weare effectively replacing the van-der-Waals interaction by an interaction potential whose value isβ ≫ Ω for nearest neighbors and 0 for atoms which are further apart (remember that Ω ≫ V2).This model is then valid provided that

1≫ Ω/β = a6Ω/C6 ≫ 1/64,i.e., Ω has to be much smaller than the nearest neighbor interaction but at the same time muchlarger than the next nearest neighbor interaction, e.g., β = 10Ω.

We can use perturbation theory to estimate the probability of transition between states cou-pled by HL, which we consider here to be a time-independent perturbation to the interactionHamiltonian. Hence, the probability of transition between states that are nearly degenerate, i.e.,belonging to the same ν-subspace, is proportional to Ω, while the transitions between subspaceswith ∣ν −ν′∣ = 1,2, energetically separated by β and 2β, respectively, occur with a probability Ω2/β.As a consequence, for sufficiently strong interaction, one can neglect the inter-subspace transitions.

The physical initial state is ∣0⟩ = ∏Lk=1 ∣P ⟩k, where no Rydberg excitation is present. This state

belongs to the subspace with ν = 0 which, as we have pointed out, we consider uncoupled to anyother state including higher number of excitation pairs. We call this the perfect blockade approachsince the time evolution takes place now only in a space spanned by states where no neighboringRydberg excitation takes place, i.e., the set of states ∣Φ⟩ such that

Hint ∣Φ⟩ = 0. (7.2)

It is valid for timescales larger than the time the system takes to perform a transition betweenadjacent ν-subspaces, i.e., such that t≪ β/Ω2.

The restriction to the ν = 0 subspace leads to a further reduction of the dimension of Hilbertspace in which the temporal evolution takes place. For example, for L = 10 the number of statesto be considered in the basis set expansion decreases from 78 to 14. This is to be compared to the1024 states which span the entire Hilbert space of the system.

7.2. Graph: general behavior of expectation values of observables

The laser Hamiltonian HL couples states in the Hilbert space whose number of excitations differby one. Moreover, due to the differences in the spatial distribution of the excitations only certainstates are connected by the laser. A convenient way to illustrate the coupling is a graph as shownin Fig. 7.2. Here vertices in the same column correspond to states that contain the same number ofRydberg atoms, and they are denoted by that number. A subscript is added when more than oneconfiguration with the same excitation number is possible. Note that, for an even (odd) number ofsites the maximal number of Rydberg atoms in one of these states is L

2(L−1

2). The way these states

are coupled is qualitatively similar for different lattice sizes L so, for simplicity reasons, we discusshere the lattice with L = 10. Starting from the initial state ∣0⟩, there are several excitations pathswith different probabilities that connect the states. The larger the amount of Rydberg atoms, themore constraints are found to allocate the next excitation. As a consequence, we encounter severalexcitation paths that do not reach the state with the maximal number of Rydberg excitations, but

7.3 Numerical results: time evolution of expectation values of observables 51

Figure 7.2.: Graph with the basis of fully-symmetric states for L = 10 in which the time evolution takesplace. In each column a subspace of a given number of Rydberg excitations is shown, see text for furtherinformation. The laser (Hamiltonian HL) couples only states belonging to adjacent subspaces. The couplingstrength (transition probability) between the individual states is encoded in the colors and line style.

end in others, e.g., the ∣4B⟩ and ∣4C⟩ states for L = 10. The features of these frustrated statesstrongly depend on the lattice size, and their amount increases as L is increased. In particular,their existence provokes small quantitative differences in the dynamics of two lattices with differentL for large times.

The previous features are reflected in the time evolution of all expectation values of the ob-servables we are going to study throughout this work. As an example, in Fig. 7.3, we show thetypical appearance of the temporal evolution of the Rydberg number nr obtained from theoreticalstudies [81, 114, 82, 115]. In this case and in general every time evolution, the dynamics can bealways divided into two different domains. At small times (t ≤ ttrans) nr rises first quadratically int showing subsequently a number of oscillations with high contrast. After this transient period theoscillations diminish and nr approaches a constant value around which it performs small amplitudefluctuations which decrease with increasing system size. This general appearance is independentof the exact details of the system, e.g., the actual arrangement of the atoms and the boundaryconditions. In experiments which are carried out in disordered gases the individual oscillationsmight not be observable since averaging over many experimental realizations also means averagingover many different spatial distributions of the atoms [83]. Each of these distributions gives rise toa different shape of the oscillating features in Fig. 7.3 and eventually the oscillations are washedout.

7.3. Numerical results: time evolution of expectation values of

observables

7.3.1. Two-sites density matrix

Let us start by introducing the reduced density matrix of two neighboring sites, since it is neededto investigate the temporal evolution of local properties such as the mean density of Rydberg atoms


0 10 20 30 40 500

5

10

t

n r

transientperiod

steady state

ttrans

Figure 7.3.: Number of Rydberg atoms as a function of time when initially no Rydberg atom is excited.One observes an initial (transient) phase with large contrast oscillations followed by a steady state. Thesefeatures do not depend on the actual details of the system, i.e., the exact atomic interaction, the atomicarrangement and the boundary conditions. The data here is shown for a ring lattice with 25 sites. The timeis given in units of the inverse collective Rabi frequency Ω−1.

or the entanglement between two adjacent sites. The density matrix of the full system is given by

ρ(t) = ∣Ψ(t)⟩⟨Ψ(t)∣. (7.3)

On can verify that the expectation value of an observable A can be obtained using the densitymatrix by ⟨Ψ(t)∣A∣Ψ(t)⟩ ≡ A(t) = Tr (ρ(t)A) ,where the trace of an operator is defined as the sum of the diagonal elements of the matrix rep-resenting the operator, or the integral if the operator is continuous. Since we are interested intwo-body properties, as the entanglement or the correlations, we are interested in the two-sites re-duced density matrix. It is obtained from the total density matrix (7.3) by performing the partialtrace over all the sites but two,

ρ(12)(t) = Tr3,4...L (ρ(t)) .Since the wavefunction ∣Ψ(t)⟩ is spanned in the subspace of fully-symmetric states, all sites areindistinguishable and we can take 1 and 2 as representative adjacent lattice sites. The basis forthe two-sites states is given by ∣PP ⟩, ∣PR⟩, ∣RP ⟩, ∣RR⟩. The restriction to the fully symmetricsubspace (indistinguishability of the two sites) imposes

⟨PR∣ρ(12)∣PR⟩ = ⟨RP ∣ρ(12)∣RP ⟩ ≡ χ⟨PP ∣ρ(12)∣PR⟩ = ⟨PP ∣ρ(12)∣RP ⟩ ≡ ζ,while the perfect blockade prevents the excitation of atoms in two neighboring sites and hence theentries ⟨ψ∣ρ(12)∣RR⟩ = ⟨RR∣ρ(12)∣ψ⟩ = 0for ∣ψ⟩ = ∣PP ⟩, ∣PR⟩, ∣RP ⟩, ∣RR⟩. As a consequence, the reduced density matrix has the particu-larly simple form:

ρ(12)(t) =⎛⎜⎜⎜⎝η ζ ζ 0ζ∗ χ ξ 0ζ∗ ξ∗ χ 00 0 0 0

⎞⎟⎟⎟⎠ , (7.4)


0.2

0.4

0.2

0.4

n k(t)

0.2

0.4

0 20 40 60 80 1000

0.2

0.4

t

0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

t

n k(t)

L=25L=10

L=10

L=15

L=20

L=25

a b

Figure 7.4.: Expectation value of the Rydberg density (7.6) versus time for a: four different ring sizes andt ≤ 100 (units of Ω−1), and b: detail of the short time evolution for L = 10 and 25. The computations havebeen performed assuming perfect blockade of the adjacent neighbor.

where η, χ, ζ and ξ are four time-dependent (complex) parameters. In particular, η and χ are real,and due to the normalization of the wavefunction, Tr1,2 ρ

(12)(t) = η + 2χ = 1. Hence, only three ofthese parameters are independent.

Since we are also interested in single-particle properties such as the density of Rydberg states,we calculate as well the single particle density matrix performing the trace over the states of sitenumber 2, yielding

ρ(1)(t) = Tr2 (ρ(12)(t)) = ( 1 − χ ζ

ζ∗ χ) . (7.5)

7.3.2. Rydberg density

The first local property under consideration is the time evolution of the expectation value of the

Rydberg density given by the Rydberg number operator in one site nk = (1/2) [1 + σ(k)z ]. By using

Eq. (7.5) one obtains this quantity in terms of one of the time-dependent entries of the densitymatrix

nk(t) = Tr(ρ(1)(t)nk) = χ, (7.6)

so the total number of Rydberg atoms evaluates to

nr(t) = Lnk(t) = Lχ.In Fig. 7.4a we show nk(t) as a function of time for different lattice sizes. In the right panel (Fig.7.4b) a magnified view of the short time dynamics is provided for L = 10 and 25.

We observe at first a steep increase, which is proportional to t2 (time in units of Ω−1), thatculminates in a pronounced peak located at t = 1.09. This peak is independent of the lattice size,as we are still witnessing the short time behavior. For much larger times, nk(t) becomes dependentof L and oscillates with a frequency f ≈ 0.48 about a mean value of nk(t) ≈ 0.26. This mean valueand also f turn out to be independent of the ring size, however, the exact shape of nk(t) stronglydepends on L. A similar result and a possible explanation of the nature of this value is givenin Ref. [81]. The time averages are performed by means of numerical integration over time in alarge enough interval t ∈ [5,200]. The lower limit of this interval is chosen large enough to avoidthe initial effects of turning on the laser. The higher one is kept shorter than the correspondingrevival time. The amplitude of the oscillations decreases considerably with increasing lattice size.This amplitude can be measured by means of the standard deviation. For L = 10, a quasi-steady


1 2 3 4 5 6 7 8 9 10 110

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Distance (k)

t

−1

−0.5

0

0.5

1

Figure 7.5.: Short time behavior of the density-density correlations g2(k, t) for L = 25. Correlations emergesuccessively during the time evolution (time in units of Ω−1). Due to the perfect blockade condition strongoscillations of g2(k, t) with a period k = 2 are observed.

state with large fluctuations characterized by a standard deviation σ(nk(t)) = 0.062 about nk(t) isestablished, while in the case of L = 25 these fluctuations become smaller σ(nk(t)) = 0.0065. Thesteady state encountered here is studied in depth in Chapter 8, where its approximate analyticalform is found.

7.3.3. Density-density correlation function

In this subsection we are going to study the time evolution of the equal-time density-density corre-lation function. This quantity measures the conditional probability of finding two simultaneouslyexcited atoms at a distance of k sites from each other normalized to the probability of uncorrelatedexcitation. It is defined as

g2(k, t) = n1n1+k(t)n1(t)n1+k(t) − 1.

This correlation function will give as a result g2(k, t) = 0 when the two sites separated by a distanceka are completely uncorrelated, and g2(k, t) > 0 (< 0) for correlation (anticorrelation) between thesites.

Figure 7.5 illustrates the initial evolution of g2(k, t) in the time interval t ∈ [0,5] for a L = 25lattice, i.e., the short-time behavior. The temporal and spatial structure can be understood byobserving the properties of the laser Hamiltonian HL which drives the excitation dynamics. Atthe beginning only a single particle at site 1 is excited. As a consequence, we observe that, due tothe perfect blockade condition, the neighboring site cannot be excited, so g2(1,0) = −1 (in general,g2(1, t) = −1 for all times). With the exception of the neighbor site, the probability of excitationof a second atom is uniform, it can occur at arbitrary position. As a consequence, there are nocorrelations for very short times (g2 has uniform zero value). The correlations emerge successivelyas time increases, starting with an augment at k = 2, the density-density correlation function forthat distance reaches a maximum at t ≈ 1.5. The high probability of finding two excitations at thedistance k = 2, i.e., a large value of g2(2, t), automatically gives rise to a decrease of g2(3, t) due toRydberg blockade. For larger times, a regular pattern of enhanced and suppressed density-densitycorrelations characterizes the dynamics. The regular pattern of the density-density correlationfunctions at short times is lost as time increases. As in the case of the Rydberg density, for longertimes g2(k, t) exhibits pronounced fluctuations that decrease with L and rapid oscillations arounda mean value.


1 2 3 4 5 6 7 8 9 10 11−1

−0.5

0

0.5

Distance (k)

g 2(k,t)

Figure 7.6.: Time-averaged density-density correlations g2(k, t) for L = 25. For distance k = 1, the value is

−1 due to the perfect blockade. g2(k, t) assumes a maximum for k = 2 (next-nearest neighbor). For largerdistances, only weak correlations are visible.

In Fig. 7.6 we show the time-averaged density-density correlation function, g2(k, t), in thestationary long time regime as a function of k, for a lattice of 25 sites. This function shows amaximum for k = 2, while for larger intersite distances it approaches the constant value 1, i.e., nocorrelations. As a consequence, we conclude that the density-density correlations are only shortranged after the initial period in which also long ranged correlations are of importance.

7.3.4. Entanglement

We study the quantum and classical correlations and the entanglement of two neighboring sites inthis system by means of the two-party correlation measure [116] and the entanglement of formation[117]. These quantities can be directly related to the entries of the reduced density matrix discussedpreviously.

Two-party correlation. The two-party correlation measure [116] is based on the trace distance[118] and it is defined as

MC (ρ(12)) = 2

3Tr∣ρ(12) − ρ(1) ⊗ ρ(2)∣, (7.7)

where ∣A∣ ≡√AA is the positive square root of AA. Its physical meaning is the distance betweenthe state ρ(12) and its reduced product state ρ(1) ⊗ ρ(2). It takes into account both the classicalcorrelation between two sites and the quantum coherence. It generalizes the classical distance in thesense that if the two operators commute then it is equal to the classical trace or Kolmogorov distancebetween the eigenvalues of ρ(12) and ρ(1) ⊗ ρ(2). We show the time evolution of this correlationmeasure in Fig. 7.7a for different sizes of the ring. Initially, for the product state ∣0⟩ = ∏L

k=1 ∣P ⟩k,there are no correlations. Analogously to the previously analyzed quantities, MC exhibits an L-independent short time behavior which here is characterized by large amplitude oscillations. It isfollowed by an L-dependent regime, whereMC presents smooth oscillations around the mean valueMC = 0.19. As expected, the amplitude of these oscillations decreases with increasing lattice size.

We are now interested in finding a classical counterpart to this correlation measure. To this end,we make use of the density matrix properties. The diagonal elements of a density matrix representthe probability of finding the corresponding configuration of the sites. For example, in ρ(12), (seeequation (7.4)), η and χ represent the probability of the two sites to be in the states ∣PP ⟩ and∣PR⟩ or ∣RP ⟩, respectively; note that the sites are indistinguishable. In the same way, the diagonal


0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

t

MC

(ρ(1

2))

L=15L=20L=25

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

t

MC

MCclass

a

b

Figure 7.7.: a: time evolution (time in units of Ω−1) of the two-party correlation measure MC (ρ(12)) forlong times and various lattice sizes. b: Short time behavior of MC (ρ(12)) and its classical counterpart forL = 25.

components of the reduced product density matrix ρ(1) ⊗ ρ(2) provides the probability of the twosites being in the corresponding product state, e.g., (1 − χ)2 for the state ∣P ⟩1 ⊗ ∣P ⟩2.

We take these diagonal elements d(12)i and d

(1⊗2)i of the matrices ρ(12) and ρ(1)⊗ρ(2), respectively,

as classical probability distributions. The Kolmogorov distance between these distributions, isdefined here as:

M classC (ρ(12)) ≡ 2

3

4

∑i=1

∣d(12)i − d(1⊗2)i ∣, (7.8)

and it provides a classical measure of the two-party correlation. In terms of the parameters of thedensity matrix this quantity is reduced to

M classC (ρ(12)) = 8

3χ2. (7.9)

The classical and the total two-party correlation functions are presented in Fig. 7.7b for t ≤ 4 and25 sites. One of the main features due to the quantum behavior of the system is the appearance ofthe two consecutive peaks ofMC at t = 0.88 and t = 1.09. Note that the classical counterpartM class

C

reproduces only the second maximum. Hence, the existence of the first one can be only justifiedby quantum arguments. Due to the absolute values in expression (7.7), two discontinuities appearin the derivative of MC around t ≈ 1.9 and t ≈ 2.3. They are, however, not observed in M class

C sinceit only depends on the single, smoothly varying, parameter of the density matrix, χ.

To get a deeper insight into the quantum effects on the correlations, the difference between thetotal two-party correlation and the classical measure is shown in Fig. 7.8 as a function of time.We have performed a fit to the local maxima of this numerical difference using an exponentialdecreasing function. The contribution of the quantum correlations loses importance as time isincreased and, at the same time, the classical dynamics starts to dominate the correlations betweentwo neighboring sites.

Concurrence and entanglement of formation. For a general state of two qubits represented bymeans of its two-particles density matrix ρ, the concurrence is given by [117]

C(ρ) =max 0, λ1 − λ2 − λ3 − λ4 , (7.10)


0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

t

MC

−M

Cclas

s

L=25Fitting exp

F(t)=0.2378 exp(−0.6767 t)

Figure 7.8.: Difference between the two-party correlation measure and its classical counterpart for L = 25.The dashed line corresponds to an exponential fit to the envelope of this difference.

where the λi are the square roots of the eigenvalues, in decreasing order, of the matrix ρρ, whereρ is the flipped matrix of the two-qubit general state ρ, i.e.,

ρ(12) = (σy ⊗ σy) (ρ(12))∗ (σy ⊗ σy) .The entanglement of formation of a state of two qubits is defined in terms of this concurrence as

E(ρ) = h⎛⎝1 +√1 −C(ρ)22

⎞⎠ ,with h(x) = −x log2 x−(1−x) log2(1−x). This quantity provides a measure of the resources neededto create a certain entangled state, and its range goes from 0 to 1.

Using the two-sites density matrix describing our system (7.4), and its corresponding flippedmatrix, we obtain the following λi:

λ1 = (χ + ∣ξ∣); λ2 = ∣χ − ∣ξ∣∣; λ3 = λ4 = 0.

These values give rise to two different regimes for the concurrence:

C (ρ(12)) = 2∣ξ∣ χ > ∣ξ∣2χ χ < ∣ξ∣ .

The first condition χ > ∣ξ∣ always holds for any size L of the lattice, so the concurrence yields

C (ρ(12)) = 2∣ξ∣. (7.11)

The time evolution of the entanglement for the lattices with sites L = 15,20 and 25 is presentedin Fig. 7.9a, and an enhancement of the behavior at short times for L = 25 is shown in Fig. 7.9b.Again, two different time domains can be distinguished. For short times, the entanglement offormation is independent of the ring size. Its maximal value, E (ρ) = 0.23, is reached at t = 0.73; fora further increase of time, E (ρ) drastically decreases, e.g., the second peak at t = 2.05 is reducedroughly by 80%. In the long time regime, the entanglement becomes weaker with E (ρ) eventuallyapproaching zero with characteristic fluctuations for each L. The amplitudes of these fluctuationsbecome smaller as L is increased.


0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

t

E(ρ

)

0 0.5 1 1.5 2 2.5 3 3.5 40

0.05

0.1

0.15

0.2

t

E(ρ

)

L=15L=20L=25

(b)

a

b

Figure 7.9.: time evolution of the entanglement of formation for a: long time and several sizes of the systemand b: short time for L = 25.7.4. Non-perfect Blockade

The above discussed phenomena have been investigated in the perfect blockade regime, which as-sumes that the energy scale associated to the van-der-Waals interaction is infinitely large comparedto the one related to the laser interaction, i.e., β ≫ Ω. Since the initial state has no Rydbergexcitations, the dynamics of the system is restricted to a small subspace including those eigenstatesof Hint with eigenvalue zero. However, for large but finite values of β/Ω, the states with ν = 0 arecoupled to those with ν > 0 and, as a consequence, these higher excitations might influence thedynamics.

In this section, we go beyond the perfect blockade approach and explore these couplings includ-ing their effect of up to order Ω/β. We thereby derive an effective Hamiltonian by dividing theeigenstates of Hint into two sets, characterized by their respective quantum number ν. The first setof states is formed by the subspace ν = 0, whereas the second one contains the rest of energeticallyhigh-lying excitations. In this framework, the Hamiltonian can be written as (with ∆ = 0)

H ≡ ( PHP PHQ

QHP QHQ) ,

where P and Q are the projectors on the subspaces with energy ν = 0 and ν > 0, respectively. Ageneral wavefunction can be decomposed as

Ψ ≡ ( PΨQΨ

) ,and, hence, the time-dependent Schrodinger equation reads:

i∂t ( PΨQΨ) = ( PHP PHQ

QHP QHQ)( PΨ

QΨ) . (7.12)

Due to the large energetic gap between the different subspaces and since the initial state is ∣0⟩ thatbelongs to the subspace with ν = 0, the transition probability to states with Eν ≠ E0 is very small.Thus, we can introduce an approximation assuming that the time variation of QΨ is very small andcan therefore be neglected, i.e., ∂t(QΨ) = 0 (adiabatic elimination of the high-lying energy levels).Hence, the equation of motion (7.12) is reduced to

i∂t(PΨ) = (PHP −PHQ(QHQ)−1QHP) (PΨ). (7.13)

7.4 Non-perfect Blockade 59

Note that, in this expression, PHP ≡ HP is the Hamiltonian within the perfect blockade regime.Whereas, the second term provides the first correction to this Hamiltonian and represents thecontribution of the couplings between the ν = 0 and ν > 0 subspaces. In practice, the subspacewith ν = 0 is only coupled to those with ν = 1and2 (see Fig. 7.1) by the Hamiltonian (6.3). As aconsequence, the Hamiltonian can be rewritten as

H ≡

⎛⎜⎜⎜⎝HP Ω01 Ω02 0Ω10 β +Ω1 Ω12 Ω1R

Ω20 Ω21 2β +Ω2 Ω2R

0 ΩR1 ΩR2 βR +ΩR

⎞⎟⎟⎟⎠ , (7.14)

where the subscript R denotes the energetic levels with ν > 2, and the Ωab represents the part ofthe laser Hamiltonian that couples the states of the subspaces with ν = a and ν′ = b, e.g., Ω01

represents the block that couples the ν = 0 and ν′ = 1 subspaces. In expression (7.14), QHQ can bedecomposed into the sum of a diagonal matrix, β, including the interaction between the subspaces,and a full matrix containing the couplings Ω,

QHQ =

⎛⎜⎝β 0 00 2β 00 0 βR

⎞⎟⎠ +⎛⎜⎝

Ω1 Ω12 Ω1R

Ω21 Ω2 Ω2R

ΩR1 ΩR2 ΩR

⎞⎟⎠ ≡ β + Ω.The inverse of this matrix can be approximated by

(QHQ)−1 = 1

β + Ω ≈ β−1 − β−1Ωβ−1 + . . . ,

where we have used the Neumann series (I − T )−1 = ∑∞n=0 T n for a square matrix T whose normsatisfies that ∥T ∥ < 1. Since β ≫ Ωab for any aand b, this condition is accomplished for T = β−1Ω.Finally, we obtain the following expression for the effective Hamiltonian

Heff =HP − Ω01Ω10

β− Ω02Ω20

2β+O(1/β2), (7.15)

where we only consider the first three terms and neglect higher order corrections.Let us now discuss the regime of validity of the approximate Hamiltonian (7.15). To this end

it is instructive to study a case in which the full Hamiltonian (6.3) is numerically tractable. Thishowever, can only be done for a small number of sites. In the absence of the laser the eigenstatesof Hamiltonian (6.3) are those of Hint, i.e., the highly degenerate ν-manifolds. As soon as the laseris turned on this degeneracy is lifted and all the ν manifolds split up. However, if β is sufficientlylarge the manifolds are still well separated. This regime is presented in Fig. 7.10 where we showa histogram of the eigenenergies (density of states) for a lattice with L = 15 and β = 20 in unitsof Ω. Since in this case the system can contain at most 15 pairs of consecutive Rydberg atoms,we observe 15 manifolds, i.e., 0 ≤ ν ≤ 15 (remember that the value ν = 14 does not exist, as weexplained at the beginning of Section 7.1). The energetic separation between the central statesof two neighboring subspaces is given by β. A magnified view of the spectral structure for thelow-lying excitations is shown in the inset of Fig. 7.10. Within the framework of the adiabaticelimination the contribution of the ν = 1 and 2 manifolds is included up to order Ω

βin the effective

Hamiltonian (7.15). The validity of this approximation is restricted to parameter regimes in whichstates belonging to different manifolds are energetically well-separated, e.g., two adjacent manifoldsmust not overlap. For L = 15, β = 20 is the minimal value needed to ensure this separation. Forlarger lattices sizes, the value of β has to be increased since with growing L the ν-manifolds containmore and more states and thus become successively broader. For example, the width of the ν = 0manifold scales proportional to L.


0 50 100 150 200 250 3000

5

10

15

20

25

30

35

40

45

Energy

−10 0 10 20 30 40 500

10

20

30

Energy

ν=0

ν=1

ν=2

β

Figure 7.10.: Histogram of all the eigenvalues (density of states) of the full Hamiltonian for a system withL = 15 and β = 20 (units of Ω). The parameters are chosen such that the individual ν-manifolds are stillrecognizable. The inset shows a magnified view of the manifolds with ν = 0,1,2, which are broadened by theinteraction with the laser.

0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

t

n k(t)

24 26 28 30

0.24

0.26

0.28

0.3

t

n k(t)

Perfect Blockadeβ=25β=35

Figure 7.11.: The Rydberg density as a function of time for L = 20 computed with the perfect blockadetreatment, and with the adiabatic elimination scheme using β = 25 and35.

We have investigated the dynamics of a ring with L = 20 sites in the framework of the adiabaticelimination using β = 25 and 35. In Figs. 7.11 and 7.12 we show the Rydberg density and adensity-density correlation function (for k = 2) and compare them with the results obtained withinthe perfect blockade approximation. The occurring deviations are small. Only minor differencesare observed at large times, and we encounter relative errors below 6.5% and 4% for β = 25 and35, respectively. The results show that the approximated inclusion of higher ν-subspaces in thedynamics does only lead to small quantitative changes in the behavior of the investigated properties.As anticipated, the deviations reduce significantly as β is increased. More qualitative differencesare expected to occur if the r−6-tail of the Rydberg-Rydberg interaction is properly accounted for.


In this chapter we have performed a numerical analysis of the laser-driven Rydberg excitationdynamics of atoms confined to a ring lattice. By exploiting the symmetry properties of the systemand employing the assumption of a perfect Rydberg blockade we were able to perform numerically


0 5 10 15 20 25 30−1

−0.5

0

0.5

1

t

g 2(2,t)

24 26 28 300.3

0.4

0.5

0.6

t

g 2(2,t)

Perfect Blockadeβ=25β=35

Figure 7.12.: Correlation function for k = 2 as a function of time for L = 20 computed with the perfectblockade treatment, and with the adiabatic elimination scheme using β = 25 and35.

exact calculations in lattices with up to L = 25 sites. Our findings show that the temporal evolutionsof the physical quantities, e.g., the Rydberg density and the density-density correlations, can bedivided into two domains. For short times, one observes an L-independent universal behavior withlarge amplitude oscillations. For longer times, the dynamics is crucially determined by the latticesize and the analyzed properties appear to assume a quasi steady state with only small temporalfluctuations. Moreover, we studied the evolution of the entanglement as well as the quantum andclassical correlations of two neighboring sites. By separating the quantum and classical part ofthe two-party correlation we showed that quantum correlations between neighboring sites decayrapidly as time passes. In addition, the entanglement between neighboring sites turned out tobe weak in the long time limit after a quick initial increase. We eventually relaxed the perfectblockade condition by taking into account higher excitation subspaces via adiabatic elimination.Propagating the initial vacuum state with the corresponding effective Hamiltonian has only smalleffect on the time evolution of the investigated observables.

In the present work we have been focusing on the dynamical properties of this system. A nextstep would be to focus on the corresponding static properties, such as eigenstates and eigenvalues.We address this topic in Chapter 9, where we characterize the many-particle eigenstates and explainhow to address them experimentally.

Most of the results presented in this chapter are published in Ref. [114].

8. Thermalization of a strongly interacting 1D

Rydberg lattice gas

The steady state shown in Fig. 7.3 is the result of a purely coherent dynamics of a closed systemand does not come about due to dissipation stemming from the coupling to an external bath. Inthis chapter we perform a closer investigation of this steady state. In particular, we study theevolution of the atomic gas in excitation number space, i.e., between subspaces of the system’sHilbert space which contain the same number of Rydberg atoms. Throughout, we employ theperfect blockade model of a Rydberg gas introduced in Section 7.1. When studying the evolutionin excitation number space under the action of the laser, we find that the strong interaction causesquasi-random couplings between regions of the Hilbert space which contain a different (and well-defined) number of Rydberg atoms. This randomness allows us to derive an effective equation forthe time evolution of the probability of being in a subspace with certain fixed excitation number.The resulting equation possesses a steady state which solely depends on the dimension of theexcitation number subspaces. A comparison to the results obtained from the numerically exactpropagation of the Schrodinger equation performed in the previous chapter shows good agreement.

8.1. Hamiltonian in excitation number space

Due to the strong interaction a formulation of the problem in terms of the physical degrees offreedom (localized atoms) seems disadvantageous. Instead, it appears more natural to base thediscussion on the graph depicted in Fig. 7.2. Initially the system is localized on the leftmostvertex of the graph, i.e., it resides in the vacuum state ∣0⟩ = ∏L

k=1 ∣P ⟩k. Once the laser is turnedon, coupling to neighboring vertices is established and the propagation of population through thegraph sets in. Eventually, the mean Rydberg number is determined by the probability ρm(t) of thesystem to be in the subspace containing m excitations:

nr(t) = nmax

∑m=0

mρm(t), (8.1)

where nmax is the next integer smaller than or equal to L/2. ρm(t) is hereby the probability densityderived from ∣Ψ(t)⟩ = exp(−iHLt) ∣0⟩ and integrated over each column of the graph. Note that theexpressions

nr(t) = ⟨Ψ(t)∣ L

∑k=1

1 + σ(k)z

2∣Ψ(t)⟩

and (8.1) are equivalent.

In order to make the excitation number spaces explicitly appear in the equations, we use thefollowing matrix representation of the wave function:

Ψ =

⎛⎜⎜⎜⎜⎜⎜⎝

⋮Ψm−1

Ψm

Ψm+1

⋮

⎞⎟⎟⎟⎟⎟⎟⎠=Ψ0 ⊕ ...⊕Ψm ⊕ ...⊕Ψnmax

.

64 Thermalization of a strongly interacting 1D Rydberg lattice gas

The vectors Ψm are the projections of the wave function onto the space with m excitations andcontain dimm components which are labeled by the indices αm = 1, ...,dimm, i.e., dimm is thenumber of possibilities of placing m Rydberg atoms on the ring that are compatible with theconstraint (7.2). The probabilities ρm are defined by

ρm =ΨmΨm. (8.2)

In this representation the Hamiltonian becomes

H =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

0 ⋱ 0 0 0⋱ 0 Cn−1,n 0 0

0 Cn−1,n 0 Cn,n+1 0

0 0 Cn,n+1 0 ⋱

0 0 0 ⋱ 0

⎞⎟⎟⎟⎟⎟⎟⎟⎠(8.3)

where the operators Cn−1,n and Cn−1,n connect the subspaces which contain n − 1 and n Rydbergatoms. Each of the subspaces contains dimn states. Hence, Cn,n+1 is a dimn × dimn+1-matrix.The block structure of the Hamiltonian (8.3) is a direct consequence of the property of the laserHamiltonian to couple only subspaces whose excitation number differs by one.

The projection operator onto the space containing n Rydberg atoms is given by ∣n⟩ ⟨n∣, and wecan define Ψn = ⟨n ∣ Ψ⟩. This allows us to rewrite Hamiltonian (8.3) more compactly as

H =nmax

∑n,m=0

∣m⟩ ⟨m∣H ∣n⟩ ⟨n∣ = nmax−1

∑n=0

[Cn,n+1 ∣n⟩ ⟨n + 1∣ + Cn,n+1 ∣n + 1⟩ ⟨n∣] .8.2. Time evolution in the excitation number subspace

8.2.1. Time evolution of the projection operators

Our goal is to obtain the time evolution of the projection operators ∣m⟩ ⟨m∣. This will eventuallyenable us to calculate the quantities ρm(t). To this end, we consider the Heisenberg equation ofmotion

∂t ∣m⟩ ⟨m∣ = i [H, ∣m⟩ ⟨m∣]which can be formally integrated to yield

∣m⟩ ⟨m∣t − ∣m⟩ ⟨m∣0 = i∫ t

0dt′ [H, ∣m⟩ ⟨m∣t′] .

For small times τ , we obtain up to second order

∣m⟩ ⟨m∣τ − ∣m⟩ ⟨m∣0 = iτ [H, ∣m⟩ ⟨m∣0] − τ22 [H, [H, ∣m⟩ ⟨m∣0]] . (8.4)

With ∣m⟩ ⟨m∣0 ≡ ∣m⟩ ⟨m∣ the first commutator evaluates to

[H, ∣m⟩ ⟨m∣] = Cm−1,m ∣m − 1⟩ ⟨m∣ + Cm,m+1 ∣m + 1⟩ ⟨m∣−Cm,m+1 ∣m⟩ ⟨m + 1∣ − Cm−1,m ∣m⟩ ⟨m − 1∣

= Cm,m−1 ∣m − 1⟩ ⟨m∣ − Cm,m−1 ∣m⟩ ⟨m − 1∣+Cm,m+1 ∣m + 1⟩ ⟨m∣ − Cm,m+1 ∣m⟩ ⟨m + 1∣ .

8.2 Time evolution in the excitation number subspace 65

20 40 60 80 100

20

40

60

80

100

120

0 1 2

20 40 60 80 100

20

40

60

80

100

0 5 10a b

Figure 8.1.: a: Representation of the matrix C5,6 for L = 20 and b: The product C5,6C6,5, that is a quasi-

diagonal matrix, which supports the assumption of the entries of the matrices Cm−1,m being uncorrelated.

where we have used Cm−1,m = Cm,m−1. The second double-commutator becomes

1

2[H, [H, ∣m⟩ ⟨m∣]] = [Cm,m+1C

m,m+1 + Cm,m−1C

m,m−1] ∣m⟩ ⟨m∣

−Cm,m−1Cm,m−1 ∣m − 1⟩ ⟨m − 1∣ − Cm,m+1Cm,m+1 ∣m + 1⟩ ⟨m + 1∣−Cm−1,mCm,m+1 ∣m − 1⟩ ⟨m + 1∣ − Cm,m+1Cm,m−1 ∣m + 1⟩ ⟨m − 1∣+12[Cm−1,m−2Cm,m−1 ∣m − 2⟩ ⟨m∣ + Cm,m−1Cm−1,m−2 ∣m⟩ ⟨m − 2∣

+Cm+1,m+2Cm,m+1 ∣m + 2⟩ ⟨m∣ + Cm,m+1Cm+1,m+2 ∣m⟩ ⟨m + 2∣] .8.2.2. Effective equation of motion for ρm

Using Eq. (8.4) and the definition ρm =ΨmΨm = ⟨Ψ ∣m⟩ ⟨m ∣ Ψ⟩ we find

ρm(τ) − ρm(0) = iτ ⟨Ψ∣ [H, ∣m⟩ ⟨m∣] ∣Ψ⟩ − τ22⟨Ψ∣ [H, [H, ∣m⟩ ⟨m∣]] ∣Ψ⟩ . (8.5)

The first commutator contains terms of the form

⟨Ψ ∣m − 1⟩ Cm,m−1 ⟨m ∣ Ψ⟩ = Ψm−1C

m,m−1Ψm

=

dimm−1

∑αm−1=1

dimm

∑βm=1

[Ψm−1]αm−1

[Cm,m−1]αm−1,βm

[Ψm]βm

= ∑αm−1βm

[Ψm−1]αm−1

[Cm,m−1]βm,αm−1[Ψm]βm

, (8.6)

where we have exploited that Cm−1,m is a real matrix.

The interaction between the Rydberg atoms manifests itself in the structure of the matricesCm−1,m, which were initially constructed in a product basis in which the single atoms constitute

the fundamental degrees of freedom. The strong interaction, however, favors collective excitations


which are complex superpositions of the single atom excitations. It is thus reasonable to assumethat the single atom degrees of freedom are so strongly mixed that the entries of Cm−1,m can beregarded as uncorrelated. Saying that the elements of a real matrix M of dimension dimf × dimc

are uncorrelated means that they accomplish

⟨MijMkl⟩ − ⟨Mij⟩⟨Mkl⟩ = κδikδjl,or, in other words,

⟨(MM )il⟩ = dimc

∑j=1

⟨MijMjl⟩ = κδil +m2,

where we have denoted the average value of the entries by m. If, as it is the case with the Cm−1,mmatrices, the average value m is much smaller than the autocorrelation coefficient κ, one sees thatthe corresponding product matrix is diagonal. In Fig. 8.1a we represent the matrix C5,6 for L = 20and the corresponding product C5,6C6,5 in Fig. 8.1b, and one can observe that in effect the entriesof these matrices can be considered uncorrelated. In this case the expression

Σ = ∑αm−1βm

[Ψm−1]αm−1

[Cm,m−1]βm,αm−1[Ψm]βm

just becomes a sum of random complex numbers. Its magnitude ∣Σ∣ can be estimated as follows:The components of the wave vector can be approximated by [Ψm]βm

≈ (dim)−1/2eiφβm , with dim =

∑nmax

n=0 dimn and φβmbeing some phase. With this we can obtain

∣Σ∣ ∼ cm,m−1

dim

RRRRRRRRRRR ∑αm−1βm

ei(φβm−φαm−1)RRRRRRRRRRR ≈

cm,m−1

dim

√dimm dimm−1,

where we have assumed that the complex numbers ei(φβm−φαm−1) are randomly (uniformly) dis-

tributed. This allows us to employ the relation ∣∑Lk=1 e

iαk ∣ ≈ √L since in case of randomly dis-tributed αk we are just dealing with a random walk in two dimensions. The constant cm,m−1

relates to the mean value of the entries of the matrix Cm,m−1.The same line of argument holds true for terms stemming from the double commutator which

are of the form

⟨Ψ ∣m⟩Cm,m+1Cm+1,m+2 ⟨m + 2 ∣ Ψ⟩ . (8.7)

Since the entries of the matrices Cm,m+1 and Cm+1,m+2 are not correlated, their product is again amatrix with randomly distributed elements. The magnitude of these terms can be estimated byemploying again the picture of the random walk in two dimensions. The modulus of (8.7) thenapproximately evaluates to cm,m+1,m+2

√dimm dimm+2/dim, where cn,m+1,m+2 is a constant related

to the mean value of the entries of Cm,m+1Cm+1,m+2.Qualitatively different, however, are the terms of the form

⟨Ψ ∣m⟩Cm,m+1Cm,m+1 ⟨m ∣ Ψ⟩ = ∑

αmβm

[Ψm]αm

[Ψm]βm∑γm+1

[Cm,m+1]αm,γm+1[Cm,m+1]βm,γm+1

,

where the matrix Cm,m+1 appears twice. Since the matrix elements of Cm,m+1 are uncorrelated onlythe diagonal elements of the matrix product yield on average a non-zero value, hence

∑γm+1

[Cm,m+1]αm,γm+1[Cm,m+1]βm,γm+1

≈ [κm,m+1]αmδαm,βm

.

Moreover, since the results cannot depend on the choice of the basis functions spanning a givenm-excitation subspace, we can say that [κm,m+1]αm

= κm,m+1 and, thus,

⟨Ψ ∣m⟩Cm,m+1Cm,m+1 ⟨m ∣ Ψ⟩ ≈ κm,m+1∑

αm

[Ψm]αm

[Ψm]αm

= κm,m+1ΨmΨm = κm,m+1ρm(0). (8.8)

8.3 The steady state 67

Hence, estimating κm,m+1 ∝ dimm+1 the modulus of these (diagonal) terms is proportional todimmdimm+1/dim.

We now return to Eq. (8.5). We neglect all terms of the form (8.6) and (8.7) and keep only thedominant ones, e.g., those which contain products of the form Cm,m+1C

m,m+1. By this we obtain

ρm(τ) − ρm(0) ≈ −τ2 [κm,m+1 + κm,m−1]ρm(0)+τ2 [κm−1,mρm−1(0) + κm+1,mρm+1(0)] .

Here we have used

∑αm

[Cm,m+1]αm,βm+1[Cm,m+1]αm,γm+1

≈ κm+1,mδβm+1,γm+1 ,

from which follows that

κm+1,m

κm,m+1

=dimm

dimm+1

.

We can thus make the ansatz κm,m+1 = umdimm+1 and κm+1,m = umdimm and find, omitting thet = 0 argument of ρm(0),

ρm(τ) − ρm ≈ −τ2 [umdimm+1 + um−1dimm−1]ρm+τ2 [um−1dimmρm−1 + umdimmρm+1] . (8.9)

This equation does not take into account coherent processes which have effectively been eliminatedby the neglect of terms of the form (8.6) and (8.7). Thus Eq. (8.9) cannot be valid for arbitrary smallvalues of τ as here certainly coherent effects dominate the evolution. Instead, the interval τ has tobe chosen sufficiently large such that terms of the form (8.8) dominate all other contributions whoseimportance diminishes due to the summation of complex numbers with random phases. Eq. (8.9) isthus a map that propagates the vector ρ(t0) by a ’coarse-grained’ timestep τ , i.e., ρ(t0)→ ρ(t0+τ).τ is thereby chosen much smaller than the typical timescale which governs the evolution of ρm(t).8.3. The steady state

The steady state is defined through

ρsteadyn (τ) − ρsteadyn = 0, (8.10)

i.e., it is a fix point of the mapping (8.9). The mapping contains the unknown coefficients um whichinclude information about how adjacent excitation subspaces are connected. Fortunately, in orderto determine the steady state their knowledge is not necessary. It is only required that um ≠ 0,which is always the case. The solution of Eq. (8.10) is given by

ρsteadyn =dimn

dim. (8.11)

Thus it is only the number of states contained in a given excitation number subspace that determinesthe steady state. It is actually possible to calculate the dimension of the subspaces with fixednumber of excitations, analytically. This is done by counting all states of the single atom productbasis that contain m excited atoms and obey condition (7.2). The result is

ρsteadym =1

dim

L

L −m ( L −mm ) with dim =nmax

∑m=0

L

L −m ( L −mm ) . (8.12)


Note, that this result encompasses all possible basis states and not only the fully symmetric oneswhich constitute the graph (7.2). It is interesting to see how ρ

steadym behaves in the limit of large L,

where we have nmax = L/2. It is convenient to introduce the variable α =m/L which is the numberof Rydberg atoms divided by the number of sites. Using Stirling’s formula we can approximate Eq.(8.12) by

ρsteadym ∝ 1√2πL

√1 − α

α(1 − 2α) ( (1 − α)1−ααα(1 − 2α)1−2α )

L

. (8.13)

This function has a very pronounced peak and, for large L, we can approximate ρsteadym (α) by aGaussian. The position of the maximum of the function is the solution of the equation

lnα + ln (1 −α) − 2 ln (1 − 2α) − 1

L(1 − 2α) + 1

2Lα(1 −α) = 0,where ln(x) is the natural logarithm. Since the number of sites L is taken to be very large, we canneglect the last two terms, and then obtain that the function (8.13) assumes its maximum at

αmax =1

2[1 − 1√

5] ≈ 0.276.

Around this peak value the squared width of ρsteadym (α) is given by

σ2α =1

5√5L

Hence, for large L the probability distribution in particle number space is strongly peaked withan overwhelming weight on αmax. The mean number of Rydberg atoms in the steady state is thusexpected to be nr = αmax × L and the fluctuation should vanish. The value of nr is slightly biggerthan the values reported in Ref. [81] and in the previous chapter of this work. As we will see inthe next section this is due to the particular choice of the initial state. In Appendix B, a morethorough derivation of the form of the steady state in performed for a blockade radius that goesbeyond the nearest neighbor site. There, the results are also compared to the ones obtained froma numerical time evolution of the system.

The distribution ρsteadym (α) contains the full statistics of the Rydberg atom number count. Stronginteractions are known to have an effect on the counting statistics leading to a sub-Poissoniandistribution [119, 120] of the Rydberg number. A measure for this is given by the Mandel Q-parameter

Q =n2r − nr2nr

− 1, (8.14)

which is negative/positive for a sub-/super-Poissonian distribution of the Rydberg atom numbercount. In the steady state we find

Qsteady =Lσ2ααmax

− 1 =√5 − 910

≈ −0.676 (8.15)

which shows the expected sub-Poissonian behavior.For the sake of completeness let us consider the case of non-interacting atoms. Here one obtains

for the probability density

ρm(t) = ( L

m) sin2L t cos2L−2m t (8.16)

and hence the probability density in excitation number space performs an oscillatory motion at alltimes.

8.4 Numerical results 69

0 5 10 15 20 250

5

10

t

n

a

0 5 10 15 20 250

10

20

t

n

b

Figure 8.2.: Temporal evolution of the system consisting of 25 atoms in particle number space. Initially allatoms are in the ground state, i.e., ρ0 = 1. a: In the interacting case (perfect blockade) the system reacheseventually a state in which the probability density localizes in excitation number space. b: This is notthe case in the absence of interactions. Here, the wave packet performs coherent oscillations with maximalamplitude. Note the different scale of the n-axis.

0 5 100

0.1

0.2

0.3

n

ρ n

L=20

0 5 100

0.1

0.2

n

ρ n

L=25

Figure 8.3.: Probability density ρn in excitation number space for L = 20 and L = 25. The green (thin)curves are snapshots taken during the interval 100 ≤ t ≤ 104. For these times the calculated Rydberg numbershows the steady state shown in Fig. 7.3. The dashed curve is obtained by taking the average over the setof snapshots. The red curve shows ρsteadyn as given by Eq. (8.10).

8.4. Numerical results

We are now going to compare the results that we obtained in the previous section to the actualdata extracted from a numerical propagation of the Schrodinger equation. In order to make thenumerical solution feasible we massively exploit the symmetry properties of the system. In allnumerical calculations we refer to the set of basis states which are invariant under cyclic shifts andreversal of the lattice sites as has been outlined in Section 6.2. Also, we operate only in a subspaceof the space which is spanned by all states being compatible with the perfect blockade condition(7.2).

8.4.1. Evolution into the steady state

Let us start by inspecting the evolution of ρn when choosing the vacuum as initial state, i.e., ρ0 = 1,and L = 25. The result is shown in Fig. 8.2a. For t ≤ 5 we observe a well-defined wave packet whichpropagates through the excitation number space performing an oscillatory motion. For longer times


0 5 100

0.1

0.2

0.3

n

ρ n

5 excitationsL=25

0 5 100

0.1

0.2

0.3

n

ρ n

12 excitationsL=25

Figure 8.4.: The same plot as in Fig. 8.3 using a state containing 5 and 12 excitations as initial state.In the former case the fluctuations around the average value are small and the agreement with ρsteadyn isremarkable. The latter shows substantial deviations from the steady state (8.11).

the amplitude of the oscillations decreases, however, the wave packet remains localized. There arestill significant fluctuations visible. In particular in the interval 15 ≤ t ≤ 25 remnants of the initialoscillations can be observed. In comparison to the non-interacting case which is shown in Fig. 8.2bthe localization in excitation number space is apparent. As expected from Eq. (8.16) here the wavepacket exhibits coherent oscillations with large amplitude.

8.4.2. The steady state and its dependence on the initial condition

We now proceed by monitoring ρn(t) over a time-interval in which the number of Rydberg atomsshows the steady state behavior - here we choose 100 ≤ t ≤ 104. The result is depicted in Fig. 8.3for two values of L, 20 and 25. The thin green curves show individual snapshots of ρn(t) taken atdifferent times. In addition, we present also the average of ρn(t) taken over the considered time-interval. The fluctuations around this average decrease significantly with increasing L. This is inaccordance with the behavior of the Rydberg number nr whose fluctuations around the mean valuealso diminish as L increases (see Chapter 7 and Ref. [114]). This supports the assumptions that inthe limit of very large L indeed a steady state with extremely little fluctuations is established. Forboth values of L shown in Fig. 8.3 a comparison to the steady state result (8.11) (thick red curve)reveals a shift of the probability distribution to smaller n. These deviations appear to stem fromthe particular choice of the initial state: The state ∣Ψ(0)⟩ = ∣0⟩ is localized at the leftmost vertexof the network. In this region of the graph the matrices Cn,n+1 are, however, not actually randomsince the ’randomness’ is caused by the interaction which has little or no effect when the numberof Rydberg atoms is only very small, i.e., n = 0,1,2.

That this ’edge effect’ appears to be indeed the cause of the deviation of the probability distri-bution from ρ

steadyn is corroborated by the data shown in Fig. 8.4. Here we present the same plot

as in Fig. 8.3 but the initial state has to be chosen from the subspace containing 5 excitations,e.g., it is located in the central region of the graph. The effect is not only a much better agreementof the data with ρ

steadyn but also a significant decrease of the fluctuations about the average of

ρn(t). This behavior is generic for initial states chosen from excitation number subspaces withlarge dimensions.

Large deviations of the steady state from Eq. (8.11) are again encountered when the initial stateis located close to the right hand edge of the graph, i.e., when its number of Rydberg atoms is closeto nmax (see Fig. 8.4). Hence, it becomes evident that Eq. (8.9) is not unconditionally valid. Itconstitutes a reliable approximation only if the initial state belongs to a particle number subspaceof sufficiently large dimension, i.e., far from the beginning and end of the graph 7.2. That it alsoworks well, when as initial state the vacuum is chosen, is not evident.

8.4 Numerical results 71

01234567891011120 1 2 3 4 5 6 7 8 9101112

0

0.1

0.2

mn

ρex nm

Figure 8.5.: Reduced density matrix in excitation number space ρex at t = 140. (L = 25, initial state ∣0⟩).Shown is the absolute value of the entries. ρex is well-approximated by a completely mixed state.

8.4.3. Reduced density matrix in excitation number space

So far we have only studied the probability density distribution in excitation number space. Furtherinsights can be gained by examining the reduced density matrix in excitation number space ρex

as this quantity eventually determines the outcome of a measurement of the number of Rydbergatoms. Its elements are defined by

ρexnm =min(dimm,dimn)

∑α=1

[Ψn ⊗Ψm]αα . (8.17)

In Fig. 8.5 we present a snapshot of ∣ρex∣ for a system of 25 sites at the time t = 140. The initialstate was ∣0⟩. We clearly observe that the entries of the main diagonal dominate the off-diagonalentries. Hence, there is negligible coherence between excitation number subspaces which have alarge dimension. Consequently, the reduced density matrix is well approximated by a classicalmixture ρex ≈ ∑nmax

n=0 ρn ∣n⟩ ⟨n∣. The strong interaction between the atoms in conjunction with thelaser driving erases the phase relation between excitation number subspaces. So tracing out alldegrees of freedom but those being relevant for the measurement of the Rydberg number, leavesus with a density matrix of a completely mixed state. This gives actually the impression that thesteady state we observe is a state with maximal entropy, since only the dimension of the excitationnumber subspaces determines the outcome of the measurement (see Eq. (8.11)). In fact we aredealing with a pure state at all times and only the particular measurement we are performing givesus the impression of observing a completely mixed state.

8.4.4. Connection with the microcanonical ensemble

Note that all the information about the steady state could as well have been obtained by consideringa microcanonical ensemble. Here, the microstates are just given by the zero energy eigenstates ∣φ⟩of the interaction Hamiltonian Hint defined through condition (7.2) and the steady state given inEq. (8.12) can be obtained directly by counting the number of these microstates. The fundamentalassumption underlying the microcanonical ensemble is that each of the microstates has equal weight.This assumption can clearly not be justified in the absence of the laser, even though all states havethe same energy. Only when the laser is present, the microstates ∣φ⟩ defined by (7.2) becomestrongly mixed and are, thus, no longer eigenstates of the system. However, they no longer possessstrictly zero energy but are rather distributed over an energy interval which is centered at zero


and whose width is proportional to LΩ. In other words, although the laser produces a widening ofthe energy window occupied by the states, it also provides the necessary ingredient that eventuallyallows the system to thermalize, the equiprobability of the microstates.

At this point we want to remark that the microcanonical prediction to the steady state (8.12)involves all zero energy eigenstates of Hint. On the other hand, the numerical results of Section 8.4involve only a subset of all accessible states, i.e., the fully symmetric set of eigenstates. The observedagreement between the two approaches suggests that the number of fully symmetric eigenstateswith a given excitation number m is proportional to dimm.


We have investigated the origin of the steady state value of the Rydberg number which is exhib-ited in a laser driven Rydberg gas after an initial transient period. Starting from Heisenberg’sequation we have derived an effective equation of motion for the probability density in excitationnumber space. This effective equation of motion which is coarse-grained in time exhibits a steadystate. When comparing this steady state to actual numerical simulations excellent agreement isfound provided that the initial state was chosen from an excitation subspace with sufficiently largedimension. In case of an initial state containing a very small/large number of excitations still asteady state is established, however, deviations from the analytical result are obtained.

We have visualized the system by a graph whose vertices are represented by eigenstates of theinteratomic interaction. Coupling between the vertices is established by the laser-atom interaction.A similar mapping was applied in Refs. [121, 122] where interacting fermions were studied bymeans of a graph. Here, a transition between localized and delocalized eigenstates has been foundto take place as a function of the interaction strength. In our system we are in the regime ofstrong interaction and the eigenstates are delocalized throughout the entire graph. The observedlocalization in excitation number space and hence also the observation of a steady state value ofnr is a purely statistical effect, owed to the strongly peaked function dimm.

It would be interesting to see whether a distribution of the Rydberg number count similar toEq. (8.11) and - more specifically - the calculated values for the mean Rydberg number andthe Mandel Q-parameter can be observed in actual experiments. Studying the shape and thetemporal evolution of the distribution should yield insights into how this steady state is establishedas the interaction strength increases. Since our simple model is not expected to be valid in higherdimension experiments with Rydberg atoms in lattices could also help here to clarify whether and,if so, how a steady state is established.

Most of the results presented here can be found in the reference [97].

9. Strong laser driving: fermionic collective

excitations

So far, we have studied our system’s time evolution in the perfect blockade regime and found that iteventually thermalizes to a steady state. This state turns out to be a strongly mixed state productof the action of the laser field. It the following chapter, we pursue a different aim: to create many-particle pure states. To do so, we make use of the same system, described by the Hamiltonian (6.3),in a different parameter regime. In particular, we choose the strong laser driving regime (Ω≫ β),and find that in this situation the Hamiltonian is exactly solvable. We characterize the arisingmany-particle states and show how to eventually excite them by just tuning the parameters of thelaser. Finally, we introduce a fluctuating Rabi frequency in our Hamiltonian and make evident thatit allows us to study fermions exposed to a disorder potential.

9.1. Constrained dynamics

Here we focus on the limit Ω ≫ β, i.e., the laser coupling is much stronger than the interactionbetween atoms. In this regime we can also assume that only nearest neighbor interaction occurs,which is justified since here Ω ≫ β ≫ V2. Thus, the Hamiltonian that describes the dynamics ofthe system reads as (7.1), i.e.,

H =L

∑k=1

[Ωσ(k)x +∆nk + βnknk+1] . (9.1)

In this regime, the first term of the Hamiltonian (9.1) is the dominant one and it is convenient tomake it diagonal by means of a rotation of the basis. This is achieved by the unitary transformation

U = ∏Lk=1 exp (−iπ4σ(k)y ) which brings σx → σz and σz → −σx. When applied to Hamiltonian (9.1),

it yields

H = U HspinU =βL

4+Hxy +H1 +H2, (9.2)

with

Hxy =

L

∑k=1

[Ωσ(k)z + β4(σ(k)+ σ

(k+1)− + σ(k)− σ

(k+1)+ )] (9.3)

H1 =∆

2

L

∑k=1

(1 − σ(k)x ) (9.4)

H2 =β

4

L

∑k=1

[(σ(k)+ σ(k+1)+ + σ(k)− σ

(k+1)− ) − 2σ(k)x ] , (9.5)

where Hxy is the famous xy-model of a spin chain with a transverse magnetic field.Let us now analyze the importance of the individual contributions of H. As we can see in Fig.

9.1, the spectrum of H decays into manifolds of states which are separated by gaps whose width is

approximately 2Ω. This is caused by the dominant first term of Hxy. The eigenstates of σ(k)z are -

in terms of the (super)atom states - given by

∣±⟩k = 1√2U [∣P ⟩k ± ∣R⟩k]

74 Strong laser driving: fermionic collective excitations

m

m+1

H1,2

Hxy

2W

2W

m+2 Hxy

Hxy

H2H1,2

Figure 9.1.: Level structure in the regime Ω≫ β and ∣∆∣≪ Ω. The spectrum splits into manifolds which can

be labeled by the quantum number m of the operator ∑k σ(k)z . For sufficiently large Ω, the coupling between

manifolds that is established only by H1 and H2 can be neglected. The (constrained) dynamics inside them-subspaces is then determined by Hxy.

with σ(k)z ∣±⟩k = ± ∣±⟩k. Thus, each of the manifolds that determine the coarse structure of the

spectrum is spanned by a set of product states that have the same number of (super)atoms in thestate ∣+⟩. In Fig. 9.1, these manifolds are denoted by m, which is the eigenvalue of the states with

respect to the operator ∑Lk=1 σ

(k)z .

The second term ofHxy conserves the total number of ∣+⟩ (super)atoms. In other words, it couplesonly states that belong to the same m-manifold and that are nearly degenerate. As a consequence,the strength of these intra-manifold couplings due to Hxy is proportional to β. Conversely, H1 andH2 couple states that belong to manifolds with different number of (super)atoms in the state ∣+⟩.In particular, H1 and the last term of H2 flip one of the (super)atoms from ∣+⟩ to ∣−⟩ or viceversa.Thus, the coupled states belong to different manifolds with ∆m = ±1, energetically separated by2Ω. The two first terms of H2 drive a similar process, flipping always two contiguous (super)atomsin the same state simultaneously, i.e., ∣++⟩→ ∣−−⟩ or ∣−−⟩→ ∣++⟩. As a result, these terms connectstates with eigenvalue m to those with m±2, which are separated roughly by 4Ω. These features arereflected in Fig. 9.1. The transition rates between m-manifolds corresponding to H1 and H2 canbe estimated by second order perturbation theory to be of the order ∆2/Ω and β2/Ω, respectively.Hence, for sufficiently strong driving Ω≫ β, their contribution can be neglected and the system’sdynamics is constrained inside the m-manifolds. As a consequence, the Hamiltonian that drives theintra-manifold dynamics, Hxy, effectively drives the dynamics of the entire system in this parameterregime. This Hamiltonian is analytically solvable, and we thus have access to the actual spectrumand eigenstates of the system. The diagonalization of this Hamiltonian relies on the so-calledJordan-Wigner transformation and a Fourier transform that we explain thoroughly in the followingparagraph [123].

9.2. Jordan-Wigner transformation on a ring

The Pauli matrices in the Hamiltonian (9.3) obey anti-commutation and commutation relationswhen they belong to the same and different sites, respectively. Thus, the algebra is neither bosonicnor fermionic. This difficulty can be overcome by the Jordan-Wigner transformation,

ck= σ(k)+

k−1

∏j=1

(−σ(j)z ) ck =k−1

∏j=1

(−σ(j)z )σ(k)− , (9.6)

which introduces the operators ckand ck that obey the canonical fermionic algebra

ci , cj = δi,j ci , cj = ci, cj = 0.

9.2 Jordan-Wigner transformation on a ring 75

After this transformation, (9.3) takes on the form

Hxy =

L

∑k=1

[2Ω(ckck − 1

2) + β

4(c

kck+1 + ck+1ck)]

− β

4(c

Lc1 + c1cL) (eiπn+ + 1) . (9.7)

Thus, the Hamiltonian has been transformed into one which describes a chain of spinless fermionswith nearest neighbor hopping. The last term of (9.7) appears due to the periodic boundaryconditions. It depends on the operator n+ = ∑L

j=1 cjcj, which counts the total number of fermions,

which is also equivalent to the number of (super)atoms in the state ∣+⟩. Thus, depending on theparity of the number of fermions of the state, Hxy reads

H(e/o)xy =

L

∑k=1

2Ω(ckck − 1

2) + β

4

L−1

∑k=1

(ckck+1 + ck+1ck)

∓β4(cLc1 + c1cL) ,

for even (e) or odd (o) parity, respectively.

These two cases can be accounted for simultaneously in a convenient way by introducing a matrixrepresentation for the fermionic operators. They are projected onto the subspaces with even andodd eigenvalue of n+ by means of the projectors Pe/o = [1 ± eiπn+] /2, with Pe + Po = 1. Since the

Hamiltonian Hxy conserves the number of fermions, i.e., [Hxy, eiπn+] = 0, it is diagonal in this

representation and can be decomposed as

Hxy = ( PeHxyPe PeHxyPo

PoHxyPe PoHxyPo) ≡ ⎛⎝ H

(e)xy 0

0 H(o)xy

⎞⎠ .We now introduce new matrix-valued creation and annihilation operators of the form

γk= ( 0 c

k

ck

0) γk = ( 0 ck

ck 0) ,

which obey the fermionic algebra provided ck and ckare fermionic operators. The Hamiltonian can

be conveniently rewritten as

Hxy = 2ΩL

∑k=1

(γkγk − 1

2) + β

4

L−1

∑k=1

(γkγk+1 + γk+1γk)

−β4(γ

Lγ1 + γ1γL) eiπn+ , (9.8)

with

eiπn+ = ( 1 00 −1 ) .

The diagonalization of the Hamiltonian (9.8) is achieved by performing the following Fouriertransform

γk=

1√L

L

∑n=1

VnkΛn γk =

1√L

L

∑n=1

VnkΛn,


with the Fourier coefficients being

Vnk =⎛⎝ e

−i 2πL(n−1/2)k 0

0 e−i2πLnk

⎞⎠ .The operators Λ

n and Λn are matrix-valued

Λn = ( 0 Peη

n

Poηn 0

) Λn = ( 0 ηnPo

ηnPe 0) ,

with ηn and ηn being fermionic creation and annihilation operators, respectively. Defining the

eigenvalue matrix ǫn as

ǫn = 2( cos [2πL(n − 1/2)] 0

0 cos [2πLn] ) ,

the diagonalized Hamiltonian (9.8) reads

Hxy = −LΩ +L

∑n=1

(2Ω + β4ǫn)Λ

nΛn. (9.9)

As we will see in the next section, the introduction of the matrix-valued fermionic operators hasthe advantage that excited states can be constructed by applying products of Λ

n to the groundstate. As a consequence, this matrix notation allows us to automatically distinguish between theodd and even fermion cases, which otherwise has to be done manually.

9.3. Many-body states

9.3.1. Fully-symmetric states

The ground state of Hamiltonian (9.9) is given by

∣G⟩ = L

∏k=1

∣−⟩kand it is fully-symmetric. Excited states that contain N fermions are in general formed by successiveapplication of the creation operator, i.e., ∣Npq...⟩ = Λ

pΛq . . . ∣G⟩. However, not all combinations will

give rise to states that belong to the fully-symmetric subset of states accessible by a time-evolutiondefined in Section 6.2.

Let us start considering the possible cases of a single-fermion excitation. For a fully-symmetricstate we require O ∣1p⟩ = OΛ

p ∣G⟩ = OΛpO ∣G⟩ = Λ

p ∣G⟩ = ∣1p⟩, i.e.,OΛ

pO ∣G⟩ = Λp ∣G⟩ ,

with O being a placeholder for X and R. After some algebra one finds that

RηpR = ei

2πLpη

L−pe

iπn+

XηpX = e−i

2πLpηpeiπc1c1 + 1√

Lc1 (eiπn+ − 1) .

Since eiπn+ ∣G⟩ = ∣G⟩ and eiπc1c1 ∣G⟩ = ∣G⟩, only the single excitation with p = L is symmetric undercyclic shifts and reversal. Hence, the only one-fermion state that can be reached by the timeevolution reads

∣1⟩ = ΛL∣G⟩ .

9.3 Many-body states 77

To have a better physical understanding of this state, it is convenient to write it in terms of theatomic operators,

∣1⟩ = 1√L

L

∑k=1

σ(k)+ ∣G⟩ .

Thus, ∣1⟩ is a spin wave or, in other words, a superatom that extends over the entire lattice. Thesestates are of interest since they can be used as a resource for single photon generation.

For the two-fermion states, we follow the same procedure and demand

OΛpΛ

qO ∣G⟩ = Λ

pΛq ∣G⟩ .

One finds that

RηpηqR = ei

2πL(p+q−1)ηL−q+1ηL−p+1

XηpηqX = e−i

2πL(p+q−1)ηpηq + ei πL√

L[e−i 2πL pηp − e−i 2πL qηq] c1 (eiπn+ − 1) ,

from where one sees that the condition p + q − 1 = L has to be accomplished. As a result, thefully-symmetric states are

∣2p⟩ = ΛpΛ

L−p+1 ∣G⟩ ,

with p = 1 . . . ⌊L/2⌋. These are entangled states formed by superpositions of two-atom excitationsin the ring with opposite momentum. This is more clearly seen by writing everything in terms ofthe Pauli matrices

∣2p⟩ = 2

iL∑k>k′

sin(2πL(p − 1/2)(k − k′))σ(k)+ σ

(k′)+ ∣G⟩ .

These states are potentially interesting for the production of photon pairs. How they can be actuallyaccessed will be discussed in Sec. 9.4.

Finally, let us illustrate how the three-fermion excitations are formed. We have

RηpηqηrR = ei 2πL (p+q+r)ηL−peiπn+ηL−qeiπn+ηL−reiπn+ = −ei 2πL (p+q+r)ηL−pηL−qηL−reiπn+ .and thus fully symmetric three-fermion states are of the form

∣3pqr⟩ = 1√2(Λ

pΛqΛ

r −Λ

L−pΛL−q

ΛL−r) ∣G⟩ , (9.10)

with p + q + r = L,2L. Writing these eigenexcitations back in terms of the spin operators yields

∣3pqr⟩ = −√2i

L3/2 ∑k>k′>k′′

∑perm(pqr)

εpqr sin [2πL(kp + k′q + k′′r)]σ(k)+ σ

(k′)+ σ

(k′′)+ ∣G⟩ ,

where εpqr is the Levi-Civita symbol. In a similar way, states with higher number of fermions areobtained.


Figure 9.2.: Spectrum of Hamiltonian H (9.2) for a lattice of L = 10 sites versus the laser driving Ω inunits of β. In the right insets, the energies of the two- and three-fermion states are shown for Ω = 10. Fivetwo-fermion and eight three-fermion eigenenergies arise as it is analytically predicted for this lattice size.

9.3.2. Energy spectrum

Now that we have analyzed the eigenstates of the system we will focus on the correspondingeigenenergies. In the course of this investigation we will also perform a comparison of the analyticresults to the ones obtained from a numerical diagonalization of the Hamiltonian (6.3). This willallow us to assess the accuracy of our analytical approach.

Let us begin with the ground state energy. From Eq. (9.9) we can read off the value

EG = −L(Ω − β4) , (9.11)

where we have included the general energy-offset βL/4 (see Eq. (9.2)). For ∆ = 0, Ω = 10, β = 1 andL = 10, the result is EG = −97.5. This is to be compared with the numerical value of −97.63 whichis obtained by diagonalizing the Hamiltonian (6.3). We find both results to be in good agreement.For the first excited state we obtain

E1 = EG + 2Ω + β2.

Using the same set of parameters, the energy of the single-fermion state is E1 = −77.0, which isvery close to the numerically exact value −77.11. The energies of higher eigenexcitations are givenby

E2p = EG + 4Ω + β cos [2πL(p − 1/2)],

with p = 1 . . . ⌊L/2⌋, for the two-fermion case and

E3pqr = EG + 6Ω + β2[cos(2π

Lp) + cos(2π

Lq) + cos(2π

Lr)] ,

with p + q + r = L,2L, for the three-fermion one. For L = 10, we obtain five and eight differenteigenenergies for the two- and three-fermion states, respectively (see insets in Fig. 9.2). In theTables 9.1 and 9.2 we perform a comparison between the analytical and the numerical results. Adifference of less than a 0.2% is observed in all cases.

The discrepancies between analytical and numerical values are mainly caused by second orderenergy shifts due to H1 and H2 (Eqs. (9.4) and (9.5)). These contributions vanish only in the

9.3 Many-body states 79

p E2p Numerical

1 -56.55 -56.642 -56.91 -56.993 -57.50 -57.584 -58.09 -58.175 -58.45 -58.54

Table 9.1.: Energies (in units of β) of the five two-fermion states ∣2p⟩ for L = 10, ∆ = 0 and Ω = 10 andcomparison with the numerically exact values.

p q r E3pqr Numerical

1 9 10 -36.19 -36.262 8 10 -36.69 -36.751 2 7 -37.10 -37.153 7 10 -37.31 -37.361 3 6 -37.65 -37.714 6 10 -37.81 -37.871 4 5 -38.00 -38.052 3 5 -38.00 -38.06

Table 9.2.: Energies (in units of β) of the eight three-fermion states ∣3pqr⟩ for L = 10, ∆ = 0 and Ω = 10 andcomparison with the numerically exact values.

limits β/Ω → 0 and ∆/Ω → 0. Here, we will calculate them for a finite ratio. There is a constantterm in H1 which is proportional to ∆ that gives rise to a global energy shift E(1) = L∆/2. Beingaware of this shift facilitates the comparison between the numerically exact and the approximateanalytical eigenvalues for ∆ ≠ 0.

Let us focus first on the ground state. H1 and H2 only couple states whose number of fermionsdiffer by one or two (Fig. 9.1). As a consequence, only ∣1⟩ and ∣2p⟩ contribute to the second ordercorrection of the energy of the ground state. It yields

E(2)G= −L ∣∆ + β∣2

8Ω + 2β −β2

4(1 + 2

L)2 ⌊L/2⌋∑

p=1

sin2 [2πL(p − 1/2)]

4Ω + β cos [2πL(p − 1/2)] . (9.12)

Analogously, we calculate the energy shift of the first excited state, ∣1⟩, due to H1 and H2. In thiscase, we have to compute the effect of ∣G⟩, ∣2p⟩ and ∣3pqr⟩. The resulting energy correction is givenby

E(2)1 =

L ∣∆ + β∣28Ω + 2β −

∣∆ + β∣2L

⌊L/2⌋∑p=1

cot2 [ πL(p − 1/2)]

2Ω + β2(2cos [2π

L(p − 1/2)] − 1) . (9.13)

Taking the parameters ∆ = 0, Ω = 10, β = 1 and L = 10, these shifts yield E(2)G= −0.14 and

E(2)1 = −0.10. The corrected energies of the ground and the single-fermion state are now EG =

E(0)G + E(1)G + E(2)G = −97.64 and E1 = E

(0)1 + E(1)1 + E(2)1 = −77.10, much closer to the numerically

exact ones of −97.63 and −77.11, respectively. We will later see that these energy corrections canbe also useful for the selective excitation of many-particle states in the lattice.

9.3.3. Correlation functions

In this subsection we are going to study the density-density correlation function of the many-particlestates. This quantity measures the conditional probability of finding two simultaneously excited


atoms at a distance x from each other normalized to the probability of uncorrelated excitation. Itis defined - for a fully-symmetric state ∣Ψ⟩ - as

g2(x,Ψ) = ⟨n1n1+x⟩Ψ⟨n1⟩2Ψ − 1,

where we have used ⟨na⟩Ψ = ⟨nb⟩Ψ for all sites. The correlation function will give g2(x,Ψ) = 0when two sites separated by a distance x are completely uncorrelated, and g2(x,Ψ) > 0 (< 0) forcorrelation (anticorrelation) between the sites.

In particular, for the case ∣Ψ⟩ = ∣2p⟩, the correlation function can be analytically calculated. In

terms of the expectation values of the spin operators, it reads g2(x,2p) = ⟨σ(1)+ σ(1+x)− + σ(1)− σ

(1+x)+ ⟩

2p.

For x = 0 we have g2(0,2p) = 1 and for x > 0 the calculation yields

g2(x,2p) = 4

L2[(L − 2x) cos [2π

L(p − 1/2)x] + 2 sin [2π

L(p − 1/2)x] cot [2π

L(p − 1/2)]] .

By inspecting this expression for the allowed values p = 1, . . . ⌊L/2⌋, some general statements canbe made:

i) Independently of the total number of sites L, there are always two ’extremal’ cases (see Fig.9.3a) which correspond to p = 1 and p = ⌊L/2⌋: For p = 1, the correlation function shows a positivemaximum at x = 1, i.e., nearest neighbor, and then decreases monotonically and smoothly with thedistance, staying always positive; for p = ⌊L/2⌋, the nearest neighbor is pronouncedly anticorrelated,the next-nearest neighbor is correlated and this pattern of correlation-anticorrelation persists withincreasing distance.

ii) For p = ⌊L/2⌋, the oscillations of g2(x,2p) are more pronounced for even L than for odd L, seeFig. 9.3a. Indeed, the ratio of the amplitudes of the correlations for x = 1 and x = ⌊L/2⌋ is,

g2(1,2L2

)g2(L2 ,2L

2

) ∼ 1g2(1,2L−1

2

)g2(L−12 ,2L−1

2

) ∼ L3,

in the even and odd cases, respectively. Also, for an even number of sites, the correlation functionsof the two extreme cases accomplish g2(x,2L

2

) = (−1)xg2(x,21), i.e., the envelope of the oscillating

function g2(x,2L2

) is given by the smoothly decreasing g2(x,21).iii) For a fixed value of p, the amplitude of the correlations decreases with increasing number of

sites as 1/L, as can be seen in Fig. 9.3b.Numerically, we have observed agreement to the analytical results shown in Fig. 9.3. As expected,

this agreement improves with a decreasing ratio β/Ω. The correlations could be directly monitoredexperimentally provided that a site-resolved detection of atoms in the ∣+⟩-state is possible. Thenext section will deal with the open question of how these correlated states can be experimentallyaccessed.

9.4. Excitation of many particle states

Our aim is to selectively excite correlated many-body states by a temporal variation of the laserparameters. Initially the atoms shall be in the product state ∣0⟩ =∏L

k=1 ∣P ⟩k and the laser shall beturned off, i.e., Ω0(0) = 0 and ∆(0) =∆0. Starting from these initial conditions, the goal is to varyΩ0(t) and ∆(t) such that at the end of the sequence, i.e., at t = tfinal, the detuning is zero andthe laser driving is much larger than the interaction (∆(tfinal) = 0 and Ω(tfinal)/β ≫ 1). This finalsituation corresponds to the right-hand side of the spectrum presented in Fig. 9.2.

Once a desired many-particle state has been populated, and due to the limited lifetime of thehighly excited levels, which is in the order of several µs (e.g., 66 µs for Rb in the 60s state), we

9.4 Excitation of many particle states 81

2 4 6 8 10 12−0.2

−0.1

0

0.1

0.2

g 2(x,2

p)

x

2 4 6 8 10 12 14x

L=24L=25

L=20L=24L=28

p=1

p= L/2

a b

Figure 9.3.: Density-density correlation functions of the ∣2p⟩ states. a: For p = 1 and p = ⌊L/2⌋, the corre-lations show completely different behavior, i.e., smoothly decreasing and strongly oscillating, respectively.These oscillations are much more pronounced for the even value of L = 24 than for the odd, L = 25. b: Themagnitude of the correlations decreases as the number of sites L is enhanced, as can be seen for L = 20,24,28.

want to map it to an stable configuration. To do so, we first turn off the laser (Ω = 0) and thenswitch on a second one whose action can be described by the Hamiltonian

Hmap = Ωs

L

∑k=1

(skrk + rksk) + β L

∑k=1

nknk+1. (9.14)

In this expression, skand sk stand for the creation and annihilation operators of an single-atom

stable storage state ∣s⟩ on site k, respectively. In the limit where the interaction is much smallerthan the Rabi frequency of this transition, i.e., Ωs ≫ β, we can neglect the second term of thisHamiltonian. Thus, performing a global π-pulse to the considered many-particle state means toperform the mapping ∣r⟩→ ∣s⟩, such that a stable configuration is achieved.

Hence, the difficulty lies in finding a ’trajectory’ or sequence (Ω0(t),∆(t)) for which at t = tfinalonly a single many-particle state is occupied. We propose two different methods to achieve thisgoal in the following paragraphs.

9.4.1. Direct trajectory

In certain cases, one can guess a trajectory (∆(t),Ω0(t)) like the ones shown in Fig. 9.4 thateventually connects ∣0⟩ with a desired eigenstate of Hxy [115, 124], but this is not always possible.The general appearance of the laser sequence strongly depends on the sign of the initial detuning∆0. In Fig. 9.4 the two possible scenarios (taking ∆0 ≠ 0) are depicted. For ∆0 < 0, the initialstate is not the ground state of the system when the laser is turned off (Ω = 0). As a consequence,this initial state suffers several avoided crossings with other levels when Ω is increased. Thus, itis not easy to find a path through the spectrum that connects it to a single desired eigenstate ofHxy, as the one shown in Fig. 9.4a. A more general framework for finding a proper trajectoryis provided by Optimal Control theory [125]. Here, the desired fidelity with which the final stateis achieved can be set and certain constraints on the trajectory can be imposed. This method issuccessfully applied to quantum information processing [126], molecular state preparation [127] andoptimization of number squeezing of an atomic gas confined to a double well potential [128]. Thecase of ∆0 > 0 will be treated thoroughly in the next subsection.


Figure 9.4.: Possible trajectories (∆(t),Ω0(t)) through the spectrum of H with ∆0 ≠ 0 (units of β). a:When ∆0 < 0, the ground state at Ω = 0 does not coincide with the initial state, ∣0⟩, and the energy of theinitial state goes through a number of avoided crossings. A possible path through them to reach the state∣523456⟩ is shown. b: If ∆0 > 0, the initial state ∣0⟩ is adiabatically connected to the ground state ∣G⟩.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

t/tfinal

Ω/Ωfinal

∆/∆0

Figure 9.5.: Shapes of the variation of the parameters of the laser Ω(t) and ∆(t) in the from given by (9.15)and (9.16), respectively.

9.4.2. Excitation from the ground state

We present here a different route to populate single many-particle states. This is accomplished intwo steps: First, one has to prepare the ground state ∣G⟩ of Hamiltonian (9.3) in the limit Ω≫ β;once the ground state is populated, the single-fermion and two-fermion many-particle states canbe accessed by means of an oscillating detuning, that gives rise to a time-dependent H1.

Step 1: Let us start by explaining how to vary the laser parameters to prepare the ground state∣G⟩. In particular, when setting ∆0 > 0, the ground state of the system at Ω = 0 coincides with theinitial state ∣0⟩. With increasing Ω, it is adiabatically connected to the ground state ∣G⟩ of Hxy

(see Fig. 9.4b). The problem that we can encounter here is that non-adiabatic transitions to otherenergy levels occur when increasing Ω, so that we do not populate only ∣G⟩ but also other states.To avoid this, we choose a large enough value of ∆0 when the laser is still turned off (Ω = 0). Thisincreases the energy gap between ∣0⟩ and other energy levels, and, as a consequence, suppressesnon-adiabatic transitions. This initial detuning can be decreased as Ω increases so that in thedesired regime, i.e., Ωfinal ≡ Ω(tfinal)≫ β, it is set to zero. As an example, we propose the following

9.4 Excitation of many particle states 83

0.50.60.70.8

0.90.95

0.99 0.990.995

t final

L=10

30 40 50 600.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.50.60.70.8

0.9

0.95

0.99

0.995

∆0

L=15

40 50

0.50.6

0.70.8

0.90.95

0.990.995

L=20

40 50

Figure 9.6.: Fidelity ∣⟨Φ(tfinal)∣G⟩∣2 when populating the ground state of Hxy from the initial state viavariation of the parameters of the laser Ω(t) and ∆(t) in the form given by (9.15) and (9.16), respectively(Ωfinal = 10 in units of β). Several initial values of the detuning and time intervals, as well as differentlattice sizes, are considered. For a fixed value of ∆0 (units of β), better fidelities are obtained for largertime intervals (units of 1/β). For a fixed time interval, there is an optimal value of ∆0 for every lattice size,around ∆0 ≈ 45.

shapes of Ω(t) and ∆(t)Ω(t) = Ωfinal sin

2 ( πt

2tfinal) (9.15)

∆(t) = ∆0 [1 − sin2 ( πt

2tfinal)] , (9.16)

that are shown in Fig. 9.5. The obtained fidelity ∣⟨Φ(tfinal)∣G⟩∣2 for different values of the initialdetuning ∆0 and time intervals tfinal is given in Fig. 9.6, where Φ(tfinal) stands for the wavefunctionof the final state (where Ωfinal = 10β). One can see that it is actually possible to populate the desiredstate with high fidelity, e.g., over 99% is achieved for all considered lattice sizes with ∆0 = 45β andtfinal = 0.9β

−1. We find that: i) the fidelity depends only weakly on the lattice size although thedimension of the Hilbert space grows exponentially with L, and ii) as expected, for a fixed value ofthe initial detuning, the fidelity increases with the increasing length of the time interval. Note thatthe timescale and the fidelity of this whole process is also limited by the lifetime of the Rydbergstate. To illustrate this, let us take the 43s state of Rb and a lattice constant of a = 3µm. We findthen that a fidelity of above 99% is achieved for tfinal = 0.27µs and a (collective) Rabi frequencyof Ω = 33.5MHz. Accounting for the limited lifetime of the 43s state (T0 ≈ 100µs), the fidelitydecreases by a factor of fL = [exp (−tfinal/T0)]L. For example, for a lattice size of L = 20 and theprevious choice of parameters, the achieved fidelity is eventually 99% × fL = 94%. This effect canbe diminished by a different choice of parameters: A change of the principal quantum number ton = 50 and a lattice constant of a = 3.5µm increase the fidelity to 97%. This is at the expense ofan increased Rabi frequency which now has to be Ω = 84.2MHz.

If there is only one atom per site, and based on the fact that ∣G⟩ =∏Lk=1 ∣−⟩k is a product state,

an alternative procedure to this adiabatic passage can be envisaged. Starting from the vacuum ∣0⟩(also a product state with every atom in ∣g⟩), we perform a global π/2-pulse to the single-atomtransition ∣g⟩ → ∣s⟩. As a result, we obtain a product state where every atom is in a superposition[∣g⟩ + i ∣s⟩] /√2. In a second step, a π-pulse with the mapping laser described by the Hamiltonian(9.14) and with Ωs ≫ β, transfers every atom to the state [∣g⟩ − ∣r⟩] /√2, i.e., we have prepared theground state ∣G⟩. It is worth remarking that this method eliminates the lifetime limitation in thisfirst stage.


Figure 9.7.: Sketch of the excitation of the single-fermion and two-fermion states by means of an oscillating(radiofrequency) detuning using a not too large value of Ω. a: In a first step, the population is transferredby a π-pulse to the single-fermion state by tuning the frequency of the detuning on resonance with the gapω = ω1. b: A second π-pulse with ω tuned to match ω2 addresses the corresponding ∣2p⟩ state, bearing inmind that β ≫ ∆osc in this step.

Step 2: Let us show now how to address the single-fermion and two-fermion states from thisground state ∣G⟩. As we explained in Section 9.1, the Hamiltonian H1, associated with the detuning,drives transitions between neighboring manifolds, i.e., ∆m = ±1, (see Fig. 9.1). We exploit thisfact and introduce an oscillating detuning of the form ∆(t) =∆osc cos (ωt). If we tune ω to coincidewith the gap between two given states, this detuning acts effectively as a laser that couples themresonantly with a Rabi frequency that is proportional to ∆osc.

Using this oscillating detuning, we want to transfer the population from the ground to the firstexcited state (Fig. 9.7a). To do so, ω is tuned to be on resonance with the corresponding energygap, i.e., ω = ω1 = E1 − EG, and by a π-pulse we populate ∣1⟩. One has to take into account thatin the limit of Ω ≫ β the energy gap between any two neighboring manifolds is equal, i.e., alsohigher lying excitations are populated. To avoid this effect and address only the ∣1⟩ state, we canchoose a not too large value of Ω. In this regime, the second order level shifts caused by H1 andH2, that are roughly given by ∆2/Ω and β2/Ω, respectively (see Section 9.3.2), become increasinglyimportant. In particular, as it is sketched in Fig. 9.7a, the gap between ∣1⟩ and any of the ∣2p⟩levels becomes more and more different from ω1 and, as a consequence, the unwanted transitionsfall out of resonance. Analogously, the same procedure could be used to address the two-fermionmany-particle states (see Fig. 9.7b). The first π-pulse resonant with the ∣G⟩ → ∣1⟩ transition, isfollowed by another π-pulse with ω tuned to coincide with the energy gap of the specific ∣1⟩→ ∣2p⟩transition, ω = ω2 = E2p −E1. The separation between neighboring ∣2p⟩ states is of the order of βand the Rabi frequency of the transition is proportional to ∆osc. As a consequence, to populateonly a single level of the two-fermion manifold, the parameters have to accomplish that β ≫ ∆osc

and, at the same time, ∆osc has to be large enough in order to perform the transfer at a timeinterval that is much shorter than the lifetime of the Rydberg state.

9.5. Disorder

So far we have assumed a constant Rabi frequency Ω. We will now consider a situation in whichit is not constant but fluctuates from site to site randomly around the mean value Ω, i.e., Ω→ Ωk,with Ωk = Ω+ δΩk being a distribution such that the average of δΩk is equal to 0. In Eq. (9.9) thefluctuating part δΩk introduces a random single particle potential for the fermions, and gives riseto the Hamiltonian of Anderson localization [129]. Hence, a lattice gas of Rydberg atoms offersthe possibility to study fermions in a disorder potential although no external atomic motion takesplace.

9.5 Disorder 85

−2 −1 0 1 20

0.2

0.4

0.6

0.8

1

(ω−2Ω)/β

I(ω

) (a

rb. u

nits

)

Figure 9.8.: Absorption profile (excitation probability of a single fermion state) for the ∣G⟩ → ∣1⟩ transitionfor three strengths of disorder, Ω/β = 10 and L = 50. The disorder strength is controlled by the fluctuationsof the Rabi frequency around the mean value, δΩk. The results are averages over 1000 realizations.

The Hamiltonian (9.3) in this situation yields

H(d)xy =

L

∑k=1

[Ωkσ(k)z + β

4(σ(k)+ σ

(k+1)− + σ(k)− σ

(k+1)+ )] ,

so that, after the Jordan-Wigner transformation, it reads

H(d)xy =Hxy +L

∑k=1

2δΩkckck,

except for an energy offset given by ∑Lk=1 δΩk. To diagonalize this Hamiltonian, we use a transfor-

mation of the type

ck=

L

∑n=1

Vnkη′n ck =

L

∑n=1

V ∗nkη′n,

such that the Hamiltonian yields

H(d)xy = −LΩ +L

∑n=1

Σnη′nη′n,

with Σn being the corresponding eigenvalues.

The disorder destroys the symmetry properties of the system and hence also the selection rulesfor transitions between many-particle states. As a consequence, there are now in general L single-fermion states (instead of 1, ∣1⟩ = ηL ∣G⟩) accessible from the state ∣G⟩ when an oscillating detuning isapplied. These new single-fermion states are given by the application of the new creation operators,that diagonalize the Hamiltonian, to the ground state, i.e.,

∣1′n⟩ = η′n ∣G⟩ = L

∑k=1

V ∗nkck∣G⟩ n = 1, . . . ,L.

Instead of a single sharp line at ω1 = E1 − EG the (averaged) absorption profile of the ∣G⟩ → ∣1⟩-transition broadens and becomes asymmetric, see Fig. 9.8. This disorder-induced line broadeningcan be detected by counting the number of ∣+⟩-atoms as a function of the excitation frequency ω.


For small values of the disorder, one can obtain an approximation to the red wing using per-turbation theory. To do so, we have to obtain the squared of the matrix element ⟨1′n∣H1 ∣G⟩,with

H1 =1

2

L

∑k=1

(1 − σ(k)x ) ,so that

1

2⟨1′n∣ L

∑k=1

(1 − σ(k)x ) ∣G⟩ = −12 ⟨1′n∣L

∑k=1

σ(k)+ ∣G⟩ = −

√L

2⟨1′n∣1⟩ ,

with ∣1⟩ being the spin wave eigenfunction of the xy-Hamiltonian without disorder. Thus, thecalculation of the intensity of absorption is reduced to obtain the overlap

⟨1′n∣1⟩ = 1√L∑k,l

V ∗nl ⟨G∣ clck ∣G⟩ = 1√L

L

∑k=1

V ∗nk =VnV(0)L ,

where Vn and V(0)n represent two sets of eigenvectors: When there is no disorder, the Hamiltonian reads Hxy = −LΩ +∑ij c

iAijcj , with

Aij =

⎧⎪⎪⎪⎨⎪⎪⎪⎩2Ω i = j

β/4 i = j + 1, j = i + 10 otherwise

,

whose diagonal form is given by D(0) = V(0)AV(0). Its eigenfunctions and eigenvalues are

given by

V(0)nk=e−i

2πLnk√L

D(0)n = 2Ω + β2cos

2π

Ln,

so that, in particular,

V(0)Lk=

1√L. When there is disorder, H

(d)xy = −LΩ + β

4 ∑ij ciA′ijcj , where A′ = A + B such that Bij =(2δΩi) δij ≡ diδij . Here, the eigenvectors that diagonalize the matrix A′ are the set Vn. We

can treat B as a small perturbation and, thus, write

Vn =V

(0)n

+ ∑m≠n

V(0)m BV

(0)n

D(0)n −D(0)m

V(0)m.

Taking these considerations into account, the overlap yields

⟨1′n∣1⟩ =V(0)n

V(0)L+ ∑

m≠n

V(0)m BV

(0)n

D(0)n −D(0)m

V(0)m

V(0)L.

Taking into account that

V(0)n

V(0)L=1

L

L

∑k=1

ei2πLnk= δnL,

we can rewrite

⟨1′n∣1⟩ = δnL + V(0)LBV

(0)n

D(0)n −D(0)

L

(1 − δnL) = δnL + ∑j,k V(0)nj

∗BjkV

(0)Lk

D(0)n −D(0)

L

(1 − δnL)= δnL + ∑

Lj=1 dje

i 2πLnj

L(D(0)n −D(0)L )(1 − δnL).

9.5 Disorder 87

The intensity of the absorption profile is given by the modulus squared of this overlap multiplied

by L/4, and they yield I(D(0)L) = L/4 and

I(D(0)n ) = L

4∣⟨1′n∣1⟩∣2 = ∑j,k djdke

i 2πLn(j−k)

4L ∣D(0)n −D(0)L∣2 =

∑j d2j +∑j≠k djdke

i 2πLn(j−k)

4L ∣D(0)n −D(0)L∣2

=∑j δΩ

2j +∑j≠k δΩjδΩke

i 2πLn(j−k)

L ∣D(0)n −D(0)L∣2 for n ≠ L, (9.17)

where dj = 2δΩj was used.

9.5.1. Quench of a superfluid

The spatial fluctuations of Ωk can, for instance, be achieved by a speckle potential or standingwaves with incommensurate frequencies [130, 131]. We present an alternative route in which thefluctuations of the Rabi frequency are caused by the fluctuating number of particles in each sitearound the mean value N0 = Ng/L with Ng being the number of total atoms in the lattice with Lsites, i.e., Nk = N0 + δNk. Thus,

Ωk = Ω0

√Nk = Ω0

√N0 + δNk ≈ Ω(1 + δNk

2N0

) ,where we have assumed that δNk ≪ N0. In particular, we start in a situation in which Ng groundstate atoms are prepared in a superfluid state of a weak lattice with L sites, i.e.,

∣SF⟩ = 1

Ng!LNg/2 [L

∑k=1

bk]Ng ∣vac⟩

where bkcreates a ground state atom at site k. Disorder is introduced by a quench of this superfluid

through a sudden increase of the depth of the lattice potential. We want to calculate the expectedvalue of the operator Nk = b

kbk for any site k, since the sites are indistinguishable. Thus, we have,

for k = L,

⟪NL⟫ ≡ ⟨SF∣ bLbL ∣SF⟩ = 1

Ng!LNg⟨vac∣ ( L

∑k=1

bk)Ng

bLbL ( L

∑k=1

bk)Ng ∣vac⟩ . (9.18)

To write the previous expression in a simpler way, we separate the term bL and bL from the sums

as followsL

∑k=1

bk=

L−1

∑k=1

bk+ b

L=√L − 1a + b

L,

where a = 1√L−1∑L−1

k=1 bk(and a) represents the creation (annihilation) operator of a boson delocal-

ized in L − 1 sites which obeys the usual bosonic commutation relations

[a, a] = 1, [a, a] = [a, a] = 0.With this, we can rewrite

( L

∑k=1

bk)Ng

= (√L − 1a + bL)Ng

=

Ng

∑l=0

(Ng

l)√L − 1Ng−l

bL

laNg−l

,


so that the quantity (9.18) can be now expressed as

⟪NL⟫ = 1

Ng!LNg

Ng

∑l,m=0

(Ng

l)(Ng

m)√L − 12Ng−l−m ⟨vac∣ blLbLbLbLm

aNg−laNg−m ∣vac⟩=

1

Ng!LNg

Ng

∑l=0

(Ng

l)2(L − 1)Ng−l ⟨vac∣ blLbLbLbLl

aNg−laNg−l ∣vac⟩=

Ng

∑l=0

(Ng

l)(L − 1)Ng−l

LNgl =

Ng

∑l=0

(Ng

l)( 1

L)l (1 − 1

L)Ng−l

l.

In summary, we can see that the average occupation of the sites (for any site k) can be writtenthen as

⟪Nk⟫ = Ng

∑n=0

P (n)n = N0,

where P (n) is the probability distribution that determines the occupation Nk, and that is thebinomial distribution with p = 1/L,

P (n) = (Ng

n)( 1

L)n (1 − 1

L)Ng−n

. (9.19)

We can also obtain the second moment of this distribution, and we find it to be

⟪N2k⟫ = N0 (N0 + 1 − 1

L) .

The same procedure can be followed to obtain the probability distribution related to the occupationof two different sites, and we find it to be given by

P (n,m) = (Ng

n)(Ng − n

m)( 1

L)n+m (1 − 2

L)Ng−n−m

, (9.20)

so the average occupation of two sites k and j with k ≠ j is

⟪NkNj⟫ = N0 (N0 − 1

L) .

With these results one can calculate that the fluctuation of the Rabi frequency δΩk = (Ω/2N0) δNk

is a random variable whose first two moments are given as

⟪δΩk⟫ = Ω

2N0

⟪Nk −N0⟫ = 0and

⟪δΩk δΩj⟫ = ( Ω

2N0

)2 ⟪(Nk −N0)(Nj −N0)⟫ = ( Ω

2N0

)2 [⟪NkNj⟫ −N20 ] = Ω2

4N0

[δkj − 1

L] . (9.21)

In this case, the perturbative approximation to the intensity profile of the transition ∣G⟩→ ∣1⟩ canbe obtained analytically. We use the fact that δΩj is a random variable, and average the intensity(9.17) over many realizations. Hence, we can see that, since the values of ⟪δΩ2

k⟫ and ⟪δΩjδΩk⟫ areindependent of the site, one can write ⟪δΩ2

k⟫ ≡ ⟪δΩ21⟫ and ⟪δΩjδΩk⟫ ≡ ⟪δΩ1δΩ1+l⟫ with l ≠ 0, and

hence

⟪ L

∑j=1

δΩ2j⟫ = L

∑j=1

⟪δΩ21⟫ = L⟪δΩ2

1⟫,


and

⟪∑j≠k

δΩjδΩkei 2πLn(j−k)⟫ =∑

j≠k

⟪δΩ1δΩ1+l⟫ei 2πL n(j−k)= ⟪δΩ1δΩ1+l⟫∑

j≠k

ei2πLn(j−k)

= −L⟪δΩ1δΩ1+l⟫.Thus, the averaged intensity yields (considering δΩk ≪ 1)

⟪I(D(0)n )⟫ = L4 ⟪∣⟨1′n∣1⟩∣2⟫ = ⟪δΩ21⟫ − ⟪δΩ1δΩ1+l⟫

4 ∣D(0)n −D(0)L ∣2 ,

so, using the result (9.21), the perturbative approximation to the profile of absorption is, in thiscase, given by

⟪I(D(0)n )⟫ = Ω2

4N0 ∣D(0)n −D(0)L∣2 ,

for n ≠ L.


In this chapter we have studied the collective excitation of a laser-driven Rydberg gas confinedto a ring lattice. We have focused on the regime in which the interaction between the highlyexcited states is much weaker than the laser field. We found that the corresponding system canbe described as a chain of spinless fermions whose dynamics is driven by the xy-model. ThisHamiltonian can be analytically solved and, by exploiting the symmetries of the system, we wereable to completely characterize the many-particle states arising. In particular, we have shown thatthe first excited state of the Hamiltonian corresponds to a spin wave or to an excitation whichis completely delocalized all over the lattice. The two-fermion states could be expressed as asuperposition of excitation pairs and an investigation of their density-density correlation functionhas been performed. We have demonstrated that the qualitative behavior of these correlationsdiffers substantially from one state to another of the same two-fermion manifold, going from asmoothly decaying function to a pronounced correlation-anticorrelation pattern. The analyticaleigenenergies of the xy-Hamiltonian were compared to the numerical exact ones of the completeHamiltonian, and excellent agreement between both results has been found. Finally, we haveinvestigated several paths for the selective excitation of the many-particle states. One of themrelies on the variation of the laser parameters with time, finding trajectories from the initial to agiven final many-body state. The other possibility we have presented makes use of an oscillatingdetuning which allows to access excitations starting from the ground state of the Hamiltonian. Ineach step, a π-pulse is performed with the frequency of the oscillation matching the energy gapbetween the involved states. Until that point we have considered an homogeneous occupation ofthe sites of the ring lattice. The situation of having a randomly fluctuating number of atoms persite would effectively lead to a disorder potential for the fermions. This implies as well a change inthe symmetry properties of the system, so that more states become accessible by a time evolution.

Most of the results presented in this chapter can be found published in Refs. [132, 133].

10. Summary and outlook

To conclude this thesis, we would like to make a brief summary of its contents. Also, and moreimportantly, we mention possible directions of future work.

In the first part, the information-theoretic measure Fisher information is studied and its ap-plications to fields that range from the statistics to quantum mechanics become apparent. First,we use the translationally invariant Fisher information as a local measure of the disorder of aquantum-mechanical system or the amount of gradient of its associated probability distribution.In particular, we focus here in the case of the D-dimensional hydrogenic atom in both position andmomentum spaces. Let us point out that this information-theoretic quantity remains to be calcu-lated for multidimensional hydrogenic Sturmians of non-spherical character (for example, parabolicor elliptic [134, 135, 136]) in the two complementary spaces.

Then, we use the Fisher information in its original form as a tool in the estimation theory,and apply it to the so-called Rakhmanov densities associated to the families of orthogonal poly-nomial that depend upon a parameter (Laguerre and Jacobi). Among the open problems in thisdirection let us first mention the computation of the parameter-based Fisher information of thegeneralized Hermite polynomials, the Bessel polynomials and the Pollaczek polynomials. A muchmore ambitious problem is the evaluation of the Fisher quantity for the general Wilson orthogonalpolynomials among others. In addition, nothing is known for discrete orthogonal polynomials. Inthis case, however, the very notion of the parameter-based Fisher information is a subtle question[137, 138].

In the second part of this thesis, a gas of ultracold laser-driven Rydberg atoms trapped in aone-dimensional ring lattice has been studied. On one hand, the case of very strong interactionshas been considered, where the numerical time evolution of the system showed the existence of aequilibrium state arising after a short period of time. The origin of this steady state product ofa purely coherent dynamics was the topic of further investigation. On the other hand, the stronglaser driving regime proved this system to open exciting perspectives for creating complex many-particle entangled states with interesting prospects for the study of disorder and the generation ofnon-classical light.

There are several natural extensions to this work that we list now here. In our considerationswe have assumed that the atoms are strongly localized, i.e., a ≫ σ. In practice there is a finitewidth of the wave-packet, caused by the uncertainty principle and finite temperature. This will leadto disorder also in the interaction energy β, which can be also treated in the present framework.Also, as we have pointed out, the main problem one has to face in this system is the limitedlifetime of the Rydberg states, which is in the order of several microseconds. One could think ofpreparing a parallel system to the one described in this work but using polar molecules [139, 140],to overcome this lifetime limitation. Eventually, an interesting extension is also the investigationof the system in two-dimensional geometries, e.g., triangular or square lattices, as well as severalrings disposed in concentric or cylindric configurations. In all these cases, the symmetries of theparticular arrangement of the sites might give rise to new interesting many-particle states and/orsignificantly affect the time evolution of these systems.

Finally, we would like to highlight a further direction of work that unifies the two parts of thisthesis. Recently, it has been investigated how Fisher information provides a sufficient condition torecognize multiparticle entanglement and, in addition, can be used as a measure of the usefulnessof pure entangled states for sub-shot noise sensitivity of linear interferometer [141, 142]. The

92 Summary and outlook

usefulness in this respect of the many-particle entangled states arising in our Rydberg gas systemin the strong laser driving regime is to be tested. A positive result would make the treated systemeven more interesting, overall given the feasibility of an experimental access to the states outlinedin Section 9.4.

11. Conclusiones

En esta seccion se describen brevemente los resultados mas relevantes de este trabajo de tesisdoctoral y se enumeran algunos problemas abiertos en este campo.

La primera parte de esta tesis esta dedicada a la investigacion de la informacion de Fisher y devarias de sus aplicaciones en estadıstica y mecanica cuantica. En primer lugar, se ha ilustrado lautilidad de la informacion de Fisher traslacionalmente invariante como medida local del desordende un sistema mecanocuantico o de la cantidad de gradiente de la distribucion de probabilidadasociada al mismo. En particular, se ha llevado a cabo este analisis en el atomo hidrogenoideD-dimensional tanto en el espacio de posiciones como en el de momentos. Una extension naturalde este trabajo serıa el calculo de esta magnitud teorico-informacional para sistemas cuanticos mascomplejos. De hecho, nuestro estudio se podrıa completar realizando un analisis similar para elcaso de las funciones Sturmianas hidrogenoides de caracter no esferico (por ejemplo, parabolicos oelıpticos [134, 135, 136]) en los dos espacios complementarios.

A continuacion, se ha utilizado la forma original de la informacion de Fisher como herramienta enla teorıa de estimacion de parametros. Se ha derivado esta magnitud para las llamadas densidadesde Rakhmanov asociadas a las familias de polinomios ortogonales que dependen de un parametro(los polinomios de Laguerre y de Jacobi). Entre los problemas abiertos en este campo merece lapena destacar el calculo de la informacion de Fisher con respecto a un parametro para los polinomiosde Hermite generalizados, de Bessel, y de Pollaczek. Realizar un estudio similar para los polinomiosortogonales de Wilson serıa un objetivo mas ambicioso debido a la complejidad computacional queconlleva. Ademas, cabe mencionar que el estudio de los polinomios ortogonales de variable discretaes un campo aun poco explorado dado que, en este caso, la mera definicion de la informacion deFisher es una cuestion sutil [137, 138].

En la segunda parte de esta tesis, se ha estudiado un gas ultrafrıo de atomos Rydberg confinado enuna red circular monodimensional y excitado mediante un campo laser. Por un lado, se ha llevado acabo un estudio numerico del regimen en el que la interaccion entre atomos Rydberg es mucho masintensa que el acoplamiento con el campo externo. En este caso, la dinamica del sistema resultaestar caracterizada por un estado de equilibrio que se alcanza tras un corto periodo de tiempo. Seha realizado una investigacion en profundidad sobre el origen de este estado estacionario que es elproducto de una dinamica puramente coherente y se ha encontrado una relacion con el conjuntomicrocanonico. Por otro lado, en el regimen contrario, dominado por la interaccion con el campolaser, se ha hallado la posibilidad de crear estados multiparticulares complejos y entrelazados, queson de gran interes para el estudio del desorden y la generacion de estados fotonicos no clasicos.

En este trabajo, hemos supuesto que los atomos estan altamente localizados en los pozos dela red. En realidad, existe una anchura asociada al paquete de ondas debida tanto al principiode incertidumbre como a la temperatura finita del sistema. Este factor trae como consecuenciaun desorden asociado en este caso a la energıa de interaccion β, que aun ha de ser estudiado enprofundidad. Por otro lado, el mayor problema al que nos enfrentamos en este sistema es la cortavida media de los estado Rydberg, del orden de los microsegundos. Una forma de evitar esteproblema es preparar un sistema similar usando moleculas diatomicas polares [139, 140]. Ademas,serıa tambien interesante considerar redes opticas bidimensionales con distintas simetrıas como,por ejemplo, triangular, cuadrada, o anillos bien concentricos o superpuestos en una configuracioncilındrica. En estos sistemas, las diferentes simetrıas podrıan afectar a la dinamica de los mismos,dando lugar a nuevos e interesantes fenomenos fısicos, ası como estados multiparticulares.

94 Conclusiones

Para terminar, nos gustarıa destacar una cuestion mas, que unifica las dos partes de esta tesis.Recientemente se ha demostrado que la informacion de Fisher es capaz de proveer una condicionsuficiente para el reconocimiento de estados multiparticulares entrelazados. Ademas, esta medidase usa como indicador cuando un estado puro entrelazado utilizado en interferometrıa lineal, ofrececomo resultado una sensibilidad por debajo del lımite del ruido inducido por el entorno debido afluctuaciones del vacio (shot-noise) [141, 142]. Merece la pena comprobar si los estados multipar-ticulares que emergen del gas de atomos Rydberg aquı investigado son utiles en este contexto. Unresultado positivo en esta direccion dotarıa al sistema tratado de aun mayor interes, sobre tododado que estos estados pueden ser excitados experimentalmente, tal y como proponemos en laSeccion 9.4.

A. Excitation to a Rydberg state by a

two-photon transition

To excite a ground-state atom to a Rydberg state, a laser field is used, so the Hamiltonian thatdescribes the process is given by the equation (5.3). In this framework, the population oscillatesfrom the ground to the Rydberg state with a frequency given by the Rabi frequency of the laser.

In practise, this transition to the Rydberg state is usually achieved by means of a two-photonprocess, i.e., by means of an intermediate state. Let us consider as an example here the excitationof the 43s state of rubidium. In this case, the initial level is the 5s, ground state of the Rb, andthe intermediate one used to achieve the Rydberg excitation is the 5p. Hence, there are two lasersinvolved in this process: one that drives the transition 5s-5p, and another one that couples 5p to43s, with detunings ∆1 and ∆2, respectively (see Fig. A.1). Thus, the Hamiltonian describing thisprocess in the basis ∣5s⟩ , ∣5p⟩ , ∣43s⟩ can be written as

H =H0 +Hl1 +Hl2 =

⎛⎜⎝ε5s 2Ω1 cosω1t 0

2Ω1 cosω1t ε5p 2Ω2 cosω2t

0 2Ω2 cosω2t ε43s

⎞⎟⎠ ,with H0 = ε5s ∣5s⟩ ⟨5s∣ + ε5p ∣5p⟩ ⟨5p∣ + ε43s ∣43s⟩ ⟨43s∣ being the atomic Hamiltonian, and Hl1 andHl2 representing the coupling of the two laser fields to the atom. The two Rabi frequencies aredefined as Ω1 ≡

E1

2⟨5s∣ z ∣5p⟩ and Ω2 ≡

E2

2⟨5p∣ z ∣43s⟩ with E1 and E2 being the amplitudes of each

of the laser fields, which are tuned to the energy gaps ω1 = ε5p − ε5s −∆1 and ω2 = ε43s − ε5p −∆2,respectively. We go to a rotating frame by means of two unitary transformations,

U1 =

⎛⎜⎝1 0 00 e−iω1t 00 0 e−iω1t

⎞⎟⎠ U2 =

⎛⎜⎝1 0 00 1 00 0 e−iω2t

⎞⎟⎠ ,such that

U = U1U2 =

⎛⎜⎝1 0 00 e−iω1t 0

0 0 e−i(ω1+ω2)t

⎞⎟⎠ .

5p

5s

(w1, W1)

(w2, W2)

D1

43sD2

Figure A.1.: Scheme of the three levels and two lasers involved in the excitation of the Rydberg 43s state ofRb.

96 Excitation to a Rydberg state by a two-photon transition

Applying this unitary transformation to the Schrodinger equation leads to an effective Hamiltonian

Heff = UHU − iU ∂

∂tU ≈⎛⎜⎝

0 Ω1 0Ω1 ∆1 Ω2

0 Ω2 ∆1 +∆2

⎞⎟⎠ ,where we have eliminated the terms oscillating with a frequency 2ω1 and 2ω2 (RWA) and putthe energy of the 5s state to zero. If we consider now that the first laser is far-detuned from thetransition 5s to 5p, i.e., that ∆1 ≫ ∣∆1 +∆2∣, we can adiabatically eliminate the intermediate 5plevel and obtain an approximate Hamiltonian for the new two-level system such that

Heff ≈1

∆1

( Ω21 Ω1Ω2

Ω1Ω2 Ω22 +∆1 (∆1 +∆2) ) .

In summary, we have transformed the three-level initial system driven by two lasers into one two-level system where the two states involved are effectively coupled by a laser with effective Rabifrequency Ω0 = Ω1Ω2/∆1 and detuning ∆ =∆1 +∆2 + (Ω2

2 −Ω21) /∆1.

B. Perfect blockade beyond the nearest neighbor

Throughout this work, we have been focused on the regime where only the nearest neighbors in thering lattice interact. Some results are presented in this appendix for the case when more sites areconsidered, in particular in the perfect blockade regime. Originally, the interaction Hamiltonianderived in Section 6.1 reads

Hint =

L

∑k=1

d

∑l=1

Vlnknk+d,

where d stands for the number of sites that separate the two Rydberg atoms, i.e., the maximalrange of the interaction is given by approximately da, with a being the lattice constant. We haveconsidered throughout this work only nearest neighbor, i.e., d = 1. When we made use of theperfect blockade approximation, we discussed that it implied that the potential is infinity insidethe considered range and 0 outside, i.e., that Ω had to accomplish

1≫Ω

V1≫

1

64.

If instead of considering only nearest neighbor interaction we assume perfect blockade over a rangegiven by d lattice sites, then the perfect blockade regime is accomplished if

1≫Ω

Vd≫

d6(d + 1)6 ,condition that gets weaker the larger d becomes, e.g., for d = 2 and d = 3 it yields 1≫ Ω/V2 ≫ 0.088and 1 ≫ Ω/V3 ≫ 0.18, respectively. Bearing this limitation in mind, we calculate numericallythe mean number of Rydberg atoms excited following the procedure of Section 7.3.2 for d ≠ 1,from where we get indication of a scaling nk(t)d ≈ (2 (d + 1))−1. We obtain now an analyticalapproximation to these mean values from the steady state arising from the derivation in Section8.3 in order to compare these results.

The steady state of the system, with a general maximal range of interactions determined by d inthe perfect blockade regime, has been shown to be given by simply the number of states containingm excitations divided by the total dimension of the Hilbert space (8.11). This can be calculatedanalytically, and the result is

ρsteady

md=

1

dim

L

L −md ( L −mdm) with dim =

nmax

∑m=0

L

L −md ( L −mdm) ,

with nmax = ⌊L/(d+1)⌋. Now, we perform a change of variable to the density of excitations α =m/L,such that

( L −mdm

) = ( L −LαdLα

) = (L −Lαd)!(Lα)! (L −Lα(d + 1))! .If now we assume that the number of sites is very large, L ≫ 1, and that the function is stronglypeaked around the maximum of α, is located around L/2(d + 1), we can use Stirling’s formula toapproximate the steady state to

ρsteadymd

∝ 1

2πL

√1 −αd

α (1 − α(d + 1)) ( (1 − αd)1−αdαα(1 −α(d + 1))1−α(d+1) )

L

.

98 Perfect blockade beyond the nearest neighbor

d αmax αnumerical σ2αL

1 0.276 0.26 0.0892 0.194 0.17 0.0503 0.151 0.12 0.033

Table B.1.: Analytical values form the derivation of the steady state and results from a numerical timeevolution are compared. We observe that the results differ more from each other the larger the range of theinteractions. In the last column, the analytical values of the squared standard deviation are presented.

This is a strongly peaked function that can be approximated by a Gaussian when L is large enough,ρsteadymd

∝ exp [g(L,α, d)]. The position of the maximum is obtained by the derivation of the functiong with respect to α and equating to zero,

1

2Lα− 1

2L(1 − αd)(1 − α(d + 1)) + lnα + d ln [1 − αd] − (d + 1) ln [1 −α(d + 1)] = 0.Since we take L to be very large, in the previous expression the two first terms can be neglected,and thus the maximum can be obtained solving the following equation

(1 −αmax(d + 1))d+1 = αmax (1 −αmaxd)d .The inverse of the second derivative of g(L,α, d) with respect to α evaluated in the maximum αmax

provides us also the squared of the standard deviation of the steady state. Again, the considerationof L≫ 1 is taken into account, and the corresponding terms are neglected, so that the result yields

σ2α =αmax(1 − αmaxd)(1 −αmax(d + 1))

L.

The comparison of these values to the numerically obtained in Section 7.3.2 is reflected in TableB.1.

C. Perturbative corrections to the xy-model

In Section 9.3.2, we gave the expressions of the second order corrections to the energies obtainedby means of the xy-model. These shifts appear due to the effect of the terms of the HamiltonianH1 and H2, and in this Appendix a thorough calculation of the expressions (9.12) and (9.13) ispresented.

In some cases it is convenient to use the expression of H1 and H2, i.e., (9.4) and (9.5), in termsof the fermionic operators c

kand ck, so we perform the Jordan-Wigner transformation (9.6) and

obtain

H1 =∆L

2− ∆

2

L

∑k=1

(ck+ ck) eiπ∑k−1

j=1 cjcj

H2 =β

4

L

∑k=1

(ckck+1− ckck+1) − β

4(c

Lc1 − cLc1) (eiπn+ + 1) − β2

L

∑k=1

(ck+ ck) eiπ∑k−1

j=1 cjcj .

The second order perturbative corrections to the energy of a general state ∣Ψ⟩ are given by

E(2)Ψ= ∑

Φ≠Ψ

∣⟨Φ∣H1 +H2 ∣Ψ⟩∣2E(0)Ψ −E(0)Φ

,

so that the problem is basically finding the expression of the corresponding matrix elements.

Let us consider first the ground state. To calculate the correction E(2)G

we need to obtain thematrix elements ⟨G∣H1 + H2 ∣1⟩ and ⟨G∣H1 + H2 ∣2n⟩ since H1 and H2 only couple states whosenumber of fermions differ by one or two. The matrix element between ∣G⟩ and ∣1⟩ reads⟨G∣H1 +H2 ∣1⟩ = −(β

2+ ∆

2) L

∑k=1

⟨G∣ ck ∣1⟩ = − 1√L(β2+ ∆

2)∑k,j

⟨G∣ ckcj ∣G⟩ = −√L2 (β +∆) ,and the matrix element between ∣G⟩ and ∣2n⟩ is

⟨G∣H1 +H2 ∣2n⟩ = −β4⟨G∣ L

∑k=1

ckck+1 − 2cLc1 ∣2n⟩= − β

4L∑p,q

ei2πL(n−1/2)(p−q) [∑

k

⟨G∣ ckck+1cpcq ∣G⟩ − ⟨G∣ 2cLc1cpcq ∣G⟩]= −β(L + 2)

4L[ei 2πL (n−1/2) − e−i 2πL (n−1/2)]

= − iβ(L + 2)2L

sin [2πL(n − 1/2)]

where we have used that

⟨G∣ ckck+1cpcq ∣G⟩ = − ⟨G∣ ckcpck+1cq ∣G⟩ + ⟨G∣ ckcq ∣G⟩ δp,k+1= − ⟨G∣ ckcp ∣G⟩ δq,k+1 + δp,k+1δq,k= δp,k+1δq,k − δq,k+1δp,k.

Plugging in these results, the Eq. (9.12) is reproduced.

100 Perturbative corrections to the xy-model

Analogously, we derive the energy shift (9.13) of the first excited state, the single-fermion ∣1⟩,due to H1 and H2. In this case, we have to calculate the effect of the states ∣G⟩, ∣2n⟩ and ∣3lmn⟩.The matrix element between ∣1⟩ and ∣2n⟩ is given by

⟨1∣H1 +H2 ∣2n⟩ = −(β2+ ∆

2) L

∑k=1

⟨1∣σ(k)− ∣2n⟩=

i (β +∆)L3/2 ∑

k,j

∑p>q

sin(2πL(n − 1/2)(p − q)) ⟨G∣σ(j)− σ

(k)− σ

(p)+ σ

(q)+ ∣G⟩

=2i (β +∆)L3/2 ∑

k>j

sin(2πL(n − 1/2)(k − j))

=i (β +∆)√

Lcot [π

L(n − 1/2)],


⟨G∣σ(j)− σ(k)− σ

(p)+ σ

(q)+ ∣G⟩ = ⟨G∣σ(j)− σ

(p)+ σ

(k)− σ

(q)+ ∣G⟩ − ⟨G∣σ(j)− σ(k)z σ

(q)+ ∣G⟩ δkp

= − ⟨G∣σ(j)− σ(p)+ σ(k)z ∣G⟩ δkq − ⟨G∣σ(k)z σ

(j)− σ

(q)+ ∣G⟩ δkp − 2 ⟨G∣σ(j)− σ

(q)+ ∣G⟩ δkpδkj

= − ⟨G∣σ(p)+ σ(j)− σ(k)z ∣G⟩ δkq + ⟨G∣σ(j)z σ(k)z ∣G⟩ δkqδjp + ⟨G∣σ(k)z σ(j)z ∣G⟩ δkpδjq

+2 ⟨G∣σ(j)z ∣G⟩ δkpδkjδjq = δkqδjp + δkpδjq − 2δkpδkjδjq.On the other hand, the matrix element between ∣1⟩ and ∣3lmn⟩ yields⟨1∣H1 +H2 ∣3lmn⟩ = −β

4

L

∑k=1

⟨1∣ ckck+1 ∣3lmn⟩ = − i√2β

4L2∑k,j

∑p,q,r

sin [2πL(pl + qm + rn)] ⟨G∣ cjckck+1cpcqcr ∣G⟩

= − i√2β

4L2∑k,j

− sin [2πL(jl + km + (k + 1)n)] + sin [2π

L(kl + jm + (k + 1)n)]

+ sin [2πL(jl + (k + 1)m + kn)] − sin [2π

L(kl + (k + 1)m + jn)]

− sin [2πL((k + 1)l + jm + kn)] + sin [2π

L((k + 1)l + km + jn)] = 0,


⟨G∣ cjckck+1cpcqcr ∣G⟩ = − ⟨G∣ cjckcpck+1cqcr ∣G⟩ + ⟨G∣ cjckcqcr ∣G⟩ δp,k+1= ⟨G∣ cjckcpcqck+1cr ∣G⟩ − ⟨G∣ cjckcpcr ∣G⟩ δq,k+1− ⟨G∣ cjcqckcr ∣G⟩ δp,k+1 + ⟨G∣ cjcr ∣G⟩ δp,k+1δq,k

= ⟨G∣ cjckcpcq ∣G⟩ δr,k+1 + ⟨G∣ cjcpckcr ∣G⟩ δq,k+1− ⟨G∣ cjcr ∣G⟩ δq,k+1δp,k − ⟨G∣ cjcq ∣G⟩ δp,k+1δr,k + δp,k+1δq,kδr,j

= − ⟨G∣ cjcpckcq ∣G⟩ δr,k+1 + ⟨G∣ cjcq ∣G⟩ δr,k+1δp,k+ ⟨G∣ cjcp ∣G⟩ δq,k+1δr,k − δq,k+1δp,kδr,j − δp,k+1δr,kδq,j + δp,k+1δq,kδr,j

= −δr,k+1δq,kδp,j + δr,k+1δp,kδq,j + δq,k+1δr,kδp,j−δq,k+1δp,kδr,j − δp,k+1δr,kδq,j + δp,k+1δq,kδr,j .

D. Numerical methods

D.1. Necklaces and bracelets

A necklace is defined to be representant (the lexicographically smallest element) of an equivalenceclass of k -ary strings under rotation. The set of all possible k -ary necklaces of length L is denotedNk(L) and the cardinality of this set is denoted by Nk(L). The enumeration formula for thenecklaces is the following:

Nk(L) = 1

L∑d∣Lφ(d)kL/d, (D.1)

where φ(m) denotes the Euler’s totient function, giving the number of numbers in the range (1,m)that are relatively prime to m.

A bracelet is the corresponding representant of an equivalence class of k -ary strings under rotationand reversal (or a necklace that is also lexicographically minimal among the circular rotations ofits reversal). The set of all k -ary bracelets is denoted Bk(L) and has cardinality Bk(L). In eachequivalence class associated with a given bracelet, there exists at most two necklaces: the braceletitself and the necklace corresponding to the reversal of the bracelet (in some cases the two may bethe same). For example, the equivalence class that contains the bracelet 0000100101 also containsthe necklace 0010000101. The cardinality Bk(L) is given by

Bk(L) = 12(Nk(L) + k+1

2kL/2) L even

12(Nk(L) + k(n+1)/2) L odd.

(D.2)

Here, we are interested in the creation of binary bracelets as they form a basis for the fully-symmetric many-particle states (with eigenvalue 1 with respect to the operations X and R intro-duced in Section 6.2). This is due to the two only possible internal states of the atoms and thespecial symmetry of our system. Numerically, all the possible necklaces of dimension L are cre-ated following the indications of the Ref. [143], where these are generated by means of a recursivealgorithm

GenNecklaces(L, t, p, a);if t > p then

if mod(L,p) = 0 then

print a;end if

else

a(t) ∶= a(t − p);GenNecklaces(L, t + 1, p, a);for j = a(t − p) to 1 do

a(t) = j;GenNecklaces(L, t + 1, t, a);

end for

end if

initialized to t = 1, p = 1 and a(0) = 0. Once the necklaces are created, the bracelets are obtainedfrom the latter ones by taking each necklace, reversing the string, and comparing with the rest of

102 Numerical methods

bracelets already stored. For example, for L = 6, there exist three possible necklaces with three 1’s,i.e., N2(6; 3) = (001011) , (001101) , (010101). In this part of the code the first necklace is storedas a bracelet, and the second and third are reversed and compared with the first one. As a result,only two bracelets arise, i.e., B2(6; 3) = (001011) , (010101).D.2. Matrix representation of the Hamiltonian

To numerically create the Hamiltonian that represents the laser coupling between the two possible(super)atom states, i.e.,

HL = ΩL

∑k=1

(rk + rk) ,one has to take into account that we are working in a subspace of the Hilbert space spanned bythe previously mentioned bracelets ∣bi⟩. The projection into the fully-symmetric subspace can be

represented by the projector P = ∑B2(L)i=1 ∣bi⟩ ⟨bi∣ such that the creation and annihilation operators

(rkand rk) in this basis accomplish

PrkP = Pr

jP and PrkP = PrjP ∀k, j.

Taking this into account, the Hamiltonian in this fully-symmetric basis can be constructed as

HL = PHLP = LP (r1 + r1)P,i.e., deriving only one of the creation or annihilation matrices in this basis. To perform this task weuse the fact that only subspaces with a number of excitations (or 1’s in the bracelets’ notation) thatdiffer by one are connected by the matrices. Let us consider the creation matrix Pr1P . The matrixelements connecting the initial ∣bi⟩ (with N − 1 excitations) with the final ∣bf ⟩ (with N excitations)states yield

⟨bi∣Pr1P ∣bf ⟩ = ⟨bi∣ r1 ∣bf ⟩ = 1√MiMf

(⟨bi1∣ + ⋅ ⋅ ⋅ + ⟨biMi∣) r1 (∣bf1⟩ + ⋅ ⋅ ⋅ + ∣bfMf

⟩)=

1√MiMf

(⟨bi1∣ r1 ∣bf1⟩ + ⋅ ⋅ ⋅ + ⟨biMi∣ r1 ∣bfMf

⟩) ,where the bracelets ∣bi⟩ and ∣bf ⟩ have been expanded in terms of theirMi andMf different equivalentstates under rotation and reversal. Numerically, these elements are obtained as follows: Two bracelets with number of excitations differing by one are taken. For example, for L = 6,

let us consider ∣bi⟩ = (000111) and ∣bf ⟩ = (001111). These two states are rotated and reversed so that all the possible equivalent configurationsare achieved. In our example,

(000111) →(001110)(011100)(111000)(110001)(100011)(000111)

(001111) →(011110)(111100)(111001)(110011)(100111)(001111)

.

D.2 Matrix representation of the Hamiltonian 103 In the set formed by the state with fewer number of excited atoms, we transform a 0 to 1 inthe first position when possible, i.e., we apply the r1 matrix,

(001110)(011100)(111000)(110001)(100011)(000111)→(101110)(111100)(100111) . The latter set of strings is compared with the one formed by the state with higher number of

excited atoms and count the number of coincidences, which for this example is two,

(101110)(111100)(100111)

(011110)(111100)(111001)(110011)(100111)(001111).

Finally, this number is divided by the square root of the multiplicity of the considered states.We obtain in our case ⟨000111∣Pr1P ∣001111⟩ = 2/√6 ⋅ 6 = 1/3.

104 Numerical methods

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